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A Combinatorial Approach to Flag Codes

In network coding, a flag code is a collection of flags, that is, sequences of nested subspaces of a vector space over a finite field. Due to its definition as the sum of the corresponding subspace distances, the flag distance parameter encloses a hidden combinatorial structure. To bring it to light, in this paper, we interpret flag distances by means of distance paths drawn in a convenient distance support. The shape of such a support allows us to create an ad hoc associated Ferrers diagram frame where we develop a combinatorial approach to flag codes by relating the possible realizations of their minimum distance to different partitions of appropriate integers. This novel viewpoint permits to establish noteworthy connections between the flag code parameters and the ones of its projected codes in terms of well known concepts coming from the classical partitions theory.


Introduction
Random network coding, firstly introduced in [1], has proven to be the most efficient way to send data across a non-coherent network with multiple sources and sinks. Despite its efficiency, it is very susceptible to error propagation and erasures. In order to overwhelm this weakness, in [12] the authors propose an algebraic approach by simply considering subspaces of F n q as codewords and subspace codes as collections of subspaces. Since this pioneering paper, much research has been made in constructing large subspace codes, and also in determining their properties. In case all the subspaces in the code have the same dimension, we speak about constant dimension codes. To have an overview of the most important works in this subject, consult [16] and the references therein.
In [14] the authors developed techniques for using flags of constant type, that is, sequences of nested subspaces of prescribed dimensions, in network coding. In this context, collections of flags are denominated flag codes and they appear as a generalization of constant dimension codes. The recent works [2,4,5,6,13] among others, show a growing interest in this topic.
If we consider all the subspaces of a given dimension of all the flags in a flag code, we obtain a constant dimension code called projected code. In the study of flag codes, a central problem is the one of unraveling to what extent is it possible to get the parameters of a flag code from the ones of its projected codes and conversely. In [3,5,6,15] this problem has been addressed for the family of flag codes attaining the maximum distance (optimum distance flag codes) whereas in [2], the authors define consistent flag codes as precisely those ones whose projected codes completely determine their parameters. In the paper at hand, we deal with this question for general full flag codes from an innovative combinatorial perspective.
When investigating the parameters of a flag code, one of the main difficulties lies in the definition of the distance between flags: it is obtained as the sum of their subspaces distances, which causes that many different combinations of them can give the same flag distance value. To succinctly represent these possible fluctuations, in [4], the authors introduce the notion of distance vector (associated to a given distance value). Here, we draw distance vectors in the distance support to obtain the so-called distance paths. This simple geometrical idea allows us to focus on the codistance of the flag code (the complement of the distance) and hence, naturally associate to a flag code different combinatorial objects coming from the classical theory of partitions that result very convenient for our purposes.
The remain of the paper is organized as follows. In Section 2 we remember some basics on partitions and Ferrers diagrams. We also recall some background on subspace codes and flag codes. In Section 3 we address a deep study on the flag distance parameter by defining the distance support of the full flag variety which allows us to graphically represent the distance path of a couple of flags. We analyze the properties of such paths and we define the new concept of codistance of a flag code. In Section 4 we translate the information that can be read in the distance support into information encoded in a combinatorial scenario. To this end, we enrich the distance support to create a Ferrers diagram frame where each distance path will be read as a set of Ferrers subdiagrams, that is, as a set of integer partitions. At the same time, we associate to each of such partitions its underlying distribution that gives a particular splitting of the corresponding codistance. In this way, we establish a one-to-one correspondence between the set of distance paths associated to a distance value and the set of splittings associated to the corresponding codistance value. Finally, in Section 5, we take advantage of the dictionary established in the previous section and, with the help of specific objects coming from the partitions world, as Durfee rectangles, we exhibit different results that precisely relate the parameters of a flag code to the ones of its projected codes. We finish the paper with some representative examples that illustrate our results, one of them giving rise to a combinatorial characterization of full flag codes of the maximum distance.
In this section we briefly recall the main definitions and results on partitions, Ferrers diagrams, subspace codes and flag codes that will be needed along the paper.

Partitions and Ferrers diagrams
Let us first fix some notation on integer partitions and their representation by Ferrers diagrams. Our basic reference related to this subject is [8].
Definition 2.1. Given a positive integer s, a partition of s is a sequence of non-increasing positive integers λ = (λ 1 , . . . , λ m ) such that λ 1 + · · · + λ m = s. Each value λ i is called a part of λ and we say that m is the length of λ.
The number of partitions, usually denoted by p(s), was determined asymptotically by Hardy and Ramanujan [11]. A remarkable expansion by Rademacher that permits calculate p(s) more accurately can be found in [9,Chapter 5].    There are other important elements naturally associated to the Ferrers diagram of a partition, as their Durfee rectangles and squares, that will be helpful for our purposes as we will see in Section 5. Let us briefly recall the precise definition.
Definition 2.5. Given the Ferrers diagram F λ of a partition λ, we call the Durfee k-rectangle associated to F λ , and denote it by D k (λ), the largest-sized rectangle within F λ with top right vertex at the corner of F λ and such that its number of columns exceeds its number of rows by k. In particular, if k = 0, the Durfee k-rectangle will be just called the Durfee square associated to F λ and simply denoted by D(λ). In [7,10], the reader can find more information on these objects.

Subspace codes and flag codes
Throughout the paper q will denote a fixed prime power and k, n two integers with 1 k < n. Consider F q the finite field with q elements and denote by G q (k, n) the Grassmannian, that is, the set of k-dimensional subspaces of F n q . The set of vector subspaces of F n q can be equipped with different metrics but, in the current paper, we will use the so-called injection distance.
Definition 2.6. The injection distance between two subspaces U , V ⊆ F n q is defined as In particular, if U , V ∈ G q (k, n), then we have Using this distance, we can define error-correcting codes in the Grassmannian as follows.
Definition 2.7. A constant dimension code C of length n and dimension k is a nonempty subset of G q (k, n). The minimum distance of C is defined as whenever |C| 2. In case |C| = 1, we put d I (C) = 0.
Remark 2.8. Another frequent metric used when working with codes whose codewords are subspaces of F n q is the subspace distance. It is given by Observe that, if U , V ∈ G q (k, n), then Hence, in the context of constant dimension codes, the injection distance and the subspace distance are equivalent metrics. Consult [16] and the references therein for more information on this class of codes.
The concept of constant dimension code can be extended when considering flags of constant type on F n q , that is, sequences of nested subspaces of F n q where the list of corresponding dimensions is fixed. The use of flags in network coding as a generalization of constant dimension codes was first proposed in [14]. Let us recall some basic background on flag codes.
In particular, if the type vector is (1, 2, . . . , n − 1), we say that F is a full flag. The subspace F i is called the i-th subspace of F.
The set of all the flags on F n q of a fixed type vector (t 1 , . . . , t r ) is said to be the flag variety F q ((t 1 , . . . , t r ), n) ⊆ G q (t 1 , n) × · · · × G q (t r , n) and, for every i = 1, . . . , r, we define the i-projection as the map The flag variety F q ((t 1 , . . . , t r ), n) can be endowed with a metric by a natural extension of the injection distance defined in (2). More precisely, given two flags Remark 2.10. Observe that the subspace distance d S defined in (4) can also be extended to the flag variety. Given F and F ′ as above, the sum of subspace distances is an equivalent distance to d f . Due to the approach followed in this paper, it is more convenient for us to choose the injection flag distance.
Definition 2.11. A flag code of type (t 1 , . . . , t r ) on F n q is a non-empty subset C ⊆ F q ((t 1 , . . . , t r ), n). Its minimum distance is given by Example 2.12. Let {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } be the canonical F q -basis of F 6 q and consider the flag code C of type (1, 3, 5) on F 6 q given by the set of flags: Remark 2.13. Note that the i-projected code C i of C is a constant dimension code in the Grassmannian G q (t i , n). At this point it is important to underline that, albeit the projected codes are constant dimension codes closely related to a flag code, they do not determine it at all; different flag codes can share the same set of projected codes. On the other hand, the cardinality of |C i | always satisfies |C i | |C|, whereas concerning the distance, we can have It suffices to see that, if C is the flag code given in Example 2.12, we have . This range of possibilities in the distance parameter behaviour comes from the flag distance definition itself; a fixed distance value can be obtained by adding different configurations of the subspaces distances. For instance, if we take flags F 2 , F 3 as in Example 2.12 and consider F 4 = ( e 3 , e 3 , e 5 , e 6 , e 2 , e 3 , e 4 , e 5 , e 6 ), In [4] the authors deal algebraically with this question by capturing such a variability with the so-called distance vectors. The distance paths defined in Section 3 are a geometrical version of such distance vectors.
In light of the previous remark, it naturally arises the problem of obtaining the parameters of a flag code from the ones of its projected codes and conversely. In Section 5 we tackle this problem with the help of new techniques based on the combinatorial objects that we will describe along Sections 3 and 4.

Flag distance and distance paths
In this section we deepen the study of the flag distance parameter describing its particular quirks from a brand-new combinatorial viewpoint. In the remain of the paper we will always work with full flags.
As said in Section 2, the flag distance defined in (6) extends the subspace distance given in (2) in the following way: the flag distance between two given flags on a vector space is exactly the sum of the distances between their subspaces. This fact implies that, contrary to what happens with subspaces distances, flag distances conceal certain complexity in the sense that a fixed value for the flag distance can be attained from different combinations of the corresponding subspace distances.
Remark 3.1. Observe that every full flag F = (F 1 , . . . , F n−1 ) of length n − 1 can be "enlarged" to the sequence of n + 1 nested subspaces of F n q given bȳ just by adding F 0 = {0} and F n = F n q . Now, for every pair of full flags F, F ′ , it clearly holds So that the distance between two full flags F, F ′ does not change if we extend them respectively toF ,F ′ . Taking this fact into account, and for technical reasons, our study of the flag distance parameter will be undertaken by using extended full flags as in (7). However, observe that, when we consider an "extended" full flag code C, two new and trivial projected codes arise: C 0 = {0} and C n = {F n q }. These codes do not give any relevant information about C. Consequently, in our study, we will just take into account the projected codes C i with 1 i n − 1, as usual.
The injection flag distance between two (extended) full flags F, F ′ on F n q is an integer that satisfies 0 d f (F, F ′ ) D n , where if n is even, This expression is a direct consequence of the possible values that the injection distance between i-dimensional subspaces of F n q can reach. For every 0 i n, let us write R(i, n) to denote the set of attainable injection subspace distances by subspaces in G q (i, n). It is clear that R(0, n) = R(n, n) = {0} and R(i, n) = {0, 1, . . . , min{i, (n − i)}}, for i ∈ {1, 2, . . . , n − 1}.
Hence, we deduce straightforwardly the next lemma.
Using this notation, for every value of 0 i n, we consider the set of points S(i, n) of Z 2 given by Definition 3.3. For every dimension 0 i n, the set S(i, n) defined in (10) is called the distance support of the Grassmannian G q (i, n).
This geometrical representation can be generalized to the full flag variety as follows.
Definition 3.4. The distance support of the full flag variety on F n q is the set Graphically, the distance support S(n) has the following representation. Remark 3.5. The reader can appreciate that two different node styles have been used in Figure 3. This is due to the fact that not every point contributes equally when we use the support to represent flag distances. On one side, crossed dots denote null distances. On the other side, there are exactly D n circle dots representing positive distances. This dichotomy will be very useful when representing distances between pairs of flags in S(n). It is clear that the i-th column of the distance support S(n) is just the distance support S(i, n) of G q (i, n). Hence, as a consequence of Lemma 3.2, the support S(n) is symmetric with respect to the vertical line x = n 2 and the columns heights grow as the dimension gets closer to n 2 . Observe that there is a remarkable difference in the distance support shape depending on the parity of n: when n is even, the value n 2 is a dimension on the type vector. In this case, S(n) has a peak in this central dimension. In contrast, when n is odd, S(n) presents a plateau on its top. The higher attainable distances in this latter case are placed at dimensions ⌊ n 2 ⌋ and ⌈ n 2 ⌉, i.e., the closest integers (from left and right, respectively) to the value n 2 .
Remark 3.6. Concerning also the distance support shape, notice that the distance support S(n − 1) can be obtained from S(n) just by removing the set of points with coordinates (i, (n − i)) whenever 2i n. These points are the ones in the "right-roof" as in the next figure. At this point, if we consider two full flags on F n q , their flag distance can be geometrically represented by means of a collection of n+1 points in the distance support S(n), each one of them in a different column S(i, n) as follows.
Definition 3.7. Given a pair of full flags F and F ′ on F n q , we define their distance path Γ(F, F ′ ) as the directed polygonal path whose vertices are the Similarly, given a full flag code, we can consider a collection of distance paths associated to it.
Notice that every distance path in S(n) starts at the point (0, 0) and arrives to (n, 0). Nevertheless, it is important to point out that not every polygonal path with vertices in the support S(n) satisfying this condition represents a potential distance path of a pair of flags. In order to characterize the polygonal paths in S(n) also being distance paths between a couple of flags in F n q , in the following result we see that, for a given pair of full flags F, F ′ on F n q , the value Theorem 3.9. Consider F, F ′ full flags on F n q and denote Proof. The proof is based on the flags nested structure. Consider full flags F and F ′ on F n q . For every 1 i < n − 1, we have The second inclusion leads to the next inequality Using this fact, and taking into account that which is equivalent to Example 3.10. In view of Theorem 3.9, the next figure illustrates how a distance path is allowed to continue once we have fixed one of its points. From some points, for instance (3, 1), we have three options to continue. On the other hand, there are just two possibilities if we fix either the point (0, 0) or the point (5, 1). Last, paths passing through (4, 3) must contain the point (5, 2). In general, given the point (i, δ i ) in S(i, n), distance paths passing through it can only come from a point At the same time, these paths can only continue through points All in all, distance paths are, graphically, oriented polygonal paths passing through points such that consecutive vertices (i, δ i ) and (i + 1, δ i+1 ) are related according to the trident rules given by (12) and (13).
Remark 3.11. Observe that distance paths defined as above are the graphic representation of the notion of the distance vector associated to a couple flags introduced in [4]. Moreover, our Theorem 3.9 is a geometric version of Theorem 3.9 in [4], for the special case of full flags. In particular, as a consequence of that result, we can assure that, given a path Γ in S(n) satisfying (12), (13) and (14), there is always a couple of full flags F, F ′ on F n q such that Γ = Γ(F, F ′ ).
Note that any distance path Γ in S(n) consists of n segments of lines with slope equal to −1, 0 or 1. Moreover, recall that every distance path starts at the origin and ends at the point (n, 0), both with null height. Hence, the number of edges with positive slope in a distance path must coincide with the one of edges with negative slope. As a result, the number of horizontal segments appearing in Γ has the same parity than n. These latter edges will play an important role in the following section. Let us define them more precisely.
Definition 3.12. A plateau of height δ in a distance path Γ is a sequence of two consecutive vertices (i, δ), (i + 1, δ) on it. In other words, if we consider two full flags F, F ′ on F n q with Γ = Γ(F, F ′ ), a plateau appears when We denote by p Γ the number of plateaus on a given distance path Γ.
Example 3.13. The distance path in red represented in Figure 5 has two plateaus of height 1. As said before, if n is an even (resp. odd) positive integer, then every distance path Γ in S(n) contains an even (resp. odd) number of plateaus. In particular, for odd values of n, distance paths must contain at least one plateau.
Remark 3.14. Let us briefly come back to the support S(n). In the following sections it will be important to compute the number of dots associated to a given distance path. Notice that in S(i, n), a point (i, δ i ) leaves exactly δ i circle dots and one crossed dot bellow it (including the point (i, δ i ) itself). Using this idea, we can compute the associated distance of a given distance path Γ by simply counting the number of circle dots on Γ or bellow Γ. Moreover, we can relate the value d Γ to the area of the polygons determined by Γ together with the abscissa axis. In Figure 5, the red path on S(7) determines a single polygon having the points (0, 0) and (6, 0) as vertices whereas the path in blue determines two of them. On the other hand, the path in Figure 7 forms a unique polygon in S (7) with vertices on the points (0, 0) and (7, 0). Theorem 3.15. Let Γ be a distance path in S(n) such that Γ determines a unique polygon P Γ with the abscissa axis. Then the flag distance d Γ is exactly the area of P Γ .
Proof. It is enough to point out that P Γ is a reticulated polygon with vertices in the integer lattice Z 2 . We start assuming that the points (0, 0) and (n, 0) are vertices of P Γ . In this case, if we write I and B to denote the set of lattice points in the interior and the boundary of P Γ , respectively, then we have that I is a set of circle dots. However, B contains n − 1 circle dots and n + 1 crossed ones. Consequently, according to Remark 3.14, it holds d Γ = |I| + n − 1.
By means of Pick's Theorem, the area of P Γ can be computed in terms of I and B as On the other hand, if Γ determines a unique polygon P Γ with vertices (i, 0) and (j, 0), for some 0 i < j n, then the result follows by interpreting P Γ as a polygon in a smaller distance support S(j −i) with the points (0, 0) and (j −i, 0) as its vertices and arguing as above.
The following corollary follows then straightforwardly: . . , P k Γ with the abscissa axis. Then the flag distance d Γ is exactly the sum of the areas of P 1 Γ , . . . , P k Γ .
By means of Proposition 3.9 and Theorem 3.15, we can remove the coordinate axes when representing flag distances in the support S(n). In fact, we just need to study paths constructed by chaining the trident moves represented in Figure 3.10, and to count how many circle dots remain in or under such paths. Of course, different distance paths can provide the same flag distance, i.e., they leave the same amount of circle dots below them or, equivalently, above them.
Next we introduce the notion of (flag) codistance as a complementary value associated to a flag distance which will be crucial in the remain of the paper.
Definition 3.17. Given a flag distance value d, i.e., an integer such that 0 d D n , we define its (injection flag) codistance as the valued = D n − d. Similarly, given a full flag code C on F n q , we define its associated codistance as Notice that both d andd provide exactly the same information since every flag distance value determines a unique codistance value and conversely. Arguing as in Remark 3.14, the codistance can be read in the distance support as follows.
Example 3.19. Take n = 7 and consider the next distance path Γ in S (7) Γ : There are D 7 = 12 circle dots in the distance support S(7). The distance path passes exactly through 6 of them (in black) and the associated polygon P Γ contains 2 black circle dots in its interior. The area of such a polygon is exactly 8 (see the picture below). Hence, the distance path in (15) represents a possible distribution to obtain the flag distance d = 8. Indeed, 8 = 1 + 2 + 1 + 1 + 2 + 1. The corresponding codistance is 12 − 8 = 4, which is, as said in Corollary 3.18, is the number of points over the path (white circle dots).

Combinatorial perspective
In this section, we introduce some combinatorial objects closely related to the distance support. To do so, we need to enrich it with an auxiliary collection of (red) points that will allow us to obtain a Ferrers diagram. With the help of such a diagram we will establish a round trip dictionary that will allows us to obtain information, in Section 5, about the distance of a flag code in terms of classical concepts related to partitions of integers.
Let us start describing the enriched version of the distance support S(n). We complete it by adding suitable auxiliary red points as in the next picture. The resultant two-colored set of points is called enriched flag distance support or just enriched support for short. We denote it byŜ(n).  (7) andŜ (8).
Recall that the silhouette of the distance support S(n) depends on the parity of n. By contrast, the enriched versionŜ(n) has always the same triangular shape. However, the position of black/red points changes depending on the parity of n. For instance, the top vertex, which has coordinates (n/2, n), is black (resp. red) when n is even (resp. odd). Remark 4.1. As stated in Remark 3.6, the distance support S(n − 1) can be obtained from S(n) by deleting the set of points in the right-roof. Similarly, the two-colored diagramŜ (7) can be constructed fromŜ(8) by performing the same operation.
In order to give a systematic and convenient construction of the two-colored enriched supportŜ(n), we proceed as follows. First, we fix the set of points in S(n) and plot them in black. Next, we consider the distance support S(n − 1) whose points we plot in red, and translate it with the vector 1 2 , 1 2 . We obtain the set S(n − 1) + 1 2 ,  Hence, using the explicit value of D n given in formula (8), we conclude that for every n 2, the enriched supportŜ(n) contains n(n−1) 2 circle dots. It also clearly contains 2n + 1 crossed dots.
As done with distance supports in the previous part, we can remove the axis and work with the two-colored enriched supportŜ(n) without specifying the coordinates of each point.

Associated Ferrers diagrams
This subsection is devoted to describe the flag distance between full flags on F n q through the concept of distance path, by using suitable Ferrers diagrams. To do so, fixed a positive integer n, we consider the enriched distance supportŜ(n) just introduced and rotate it around the point (n, 0) it as in the next figure.   There are some partitions of positive integers r n(n − 1)/2 that can be represented by Ferrers diagrams contained in the Ferrers diagram frame (then in the primary one). The following definition precises this idea. Due to technical reasons, we also consider the empty Ferrers subdiagram F 0 , associated to the null embedded partition λ = (0) (see the second diagram in Figure 11).
Observe that these embedded partitions are exactly those ones whose associated Ferrers diagram fits in FF(n). With this notation, we conclude directly the next result. (1) F λ is a Ferrers subdiagram of FF(n).
At this point, and with the purpose of connecting these combinatorial objects with our study on the flag distance, we define a special class of polygonal paths in the primary Ferrers diagram FF(n), closely related to the set of Ferrers subdiagrams in FF(n).
Definition 4.8. A staircase path Σ on FF(n) is just a polygonal directed path whose vertices are dots of FF(n) (crossed or circle ones) such that: • it starts (resp. ends) at the highest (resp. lowest) crossed black point and • its directed edges are either vertical segments straight down or horizontal segments from left to right.
Remark 4.9. Observe that, since black and red dots are interspersed in FF(n), staircase paths travel along the diagram alternating black and red points. Even more, every staircase path contains exactly n + 1 black dots and n red ones. Moreover, since staircase paths cannot go neither up nor to the left, the collection of points that remains at right of any staircase path satisfy the next property: the number of dots at a given row is always upper bounded by the number of dots at the previous one, that is, any staircase path is the "silhouette" of a Ferrers diagram.  Given Σ a staircase path in FF(n), the set of points to the right of Σ forms a set F(Σ) that is a Ferrers subdiagram of FF(n). Conversely, a Ferrers subdiagram F of FF(n) determines a unique staircase path Σ(F) in FF(n). In this situation we say that Σ (resp. Σ(F)) is the silhouette of the Ferrers subdiagram F(Σ) (resp. F).
Proof. Consider a staircase path Σ in FF(n). Giving this path is equivalent to provide a sequence of integers λ = (λ 1 , . . . , λ n−1 ) such that, for every 1 i n − 1, the value 0 λ i n − i counts the number of circle dots in the i-th row of FF(n) that remain to the right of Σ. As said in Remark 4.9, these values satisfy λ i λ i+1 0, for every 1 i n − 2. If the sequence λ does not contain zeros, or equivalently λ n−1 = 0, then it is a partition and we have that F(Σ) = F λ . On the other hand, if λ 1 = 0, then F(Σ) = F 0 . This case corresponds to the staircase that flows horizontally up to the corner of FF(n) and then comes down vertically as in the second diagram in Figure 11. Last, if λ = (λ 1 , . . . , λ i , 0, . . . , 0), with λ i = 0, then F(Σ) = F λ ′ with λ ′ = (λ 1 , . . . , λ i ).
The previous result can be read in terms of embedded partitions as follows.

Coming back to the flag distance
In the previous subsection, we associated a couple of Ferrers diagrams to the distance support by adding a suitable collection of auxiliary red points. In that scenario, the set of staircase paths are characterized by using embedded partitions of integers. Once established these combinatorial tools, to re-connect them with our distance paths, we need to remove the auxiliary red structure somehow. This removing process will allows us describe distance paths associated to a flag distance value d in terms of appropriate summand distributions of the corresponding codistance valued. Let us explain this in more detail.
Our first objective is to consistently retrieve distance paths from staircase paths. Recall that, as stated in Remark 4.9, black and red dots alternate in a staircase path. Hence, in order to study distance paths obtained by removing red dots, it is important to observe how two consecutive black dots can be connected when the intermediate red point that appears between them in a staircase path is eliminated. There are four admissible local movements for this operation: → → → → Figure 12: Movements to locally recover a distance path by removing a red dot.
Remark 4.12. The four motion patterns described in Figure 12 are also valid when they involve crossed points.
With these four movements in mind, we do the following. We consider all the possible sequences of black-red-black points that are parts of staircase paths passing through a given black dot. We remove the red intermediate point and apply the corresponding movement in Figure 12 and get just three possibilities, labelled as 1, 2 and 3 in Figure 13, starting from the given black dot. Observe that this figure corresponds to the rotation of the trident pattern exposed in Figure 3.10 and obtained in Proposition 3.9. As a consequence, we have straightforwardly the next result.
Proposition 4.13. The polygonal path Γ(Σ) obtained after removing the red dots of any staircase path Σ in FF(n) by applying the movements in Figure 12 is a distance path in the distance support S(n) (after rotation). We call it the skeleton distance path associated to the staircase path Σ.
It is clear that different staircase paths in FF(n) can share the same associated skeleton distance path in S(n) (after rotation). Hence, we can define an equivalence relation on the set of staircase paths. Definition 4.14. We say that two staircase paths Σ and Σ ′ in FF(n) are distance-equivalent if they have the same associated skeleton distance path, i.e., Γ(Σ) = Γ(Σ ′ ). Given a distance path Γ in S(n), we denote the set of distanceequivalent staircase paths with Γ as their skeleton distance path as Σ(Γ).
Example 4.15. In Figure 14, we can see a distance path Γ in S(8) and the same path represented in the enriched supportŜ (8). Below, in Figure 15, we represent in red the four possible staircase paths in Σ(Γ), i.e., those staircase paths having Γ as their associated skeleton distance path.   Proof. Let Γ a the distance path in the distance support S(n). Consider an arbitrary edge e of Γ, connecting two consecutive vertices (i, δ i ) and (i+ 1, δ i+1 ), for some 0 i n − 1, and assume that Σ is a staircase path passing through these two black points too. Let us study the possibilities for the red point in Σ connecting these two black ones. By virtue of Proposition 3.9, we know that e has slope either −1, 0 or 1. If it has slope equal to −1 (resp. 1), then after rotation we obtain a vertical (resp. horizontal) segment that already determines the unique red point connecting the starting vertices, as we can see in the next figure.
→ Figure 16: Segments of slope 1 (in black) and −1 (in green) in S(n) (left). The same segments seen in FF(n) (right).
On the other hand, if e has slope equal to 0, i.e., it is a plateau, then it is transformed (after rotation) into a segment with slope −1 in FF(n). It can be replaced by two sequences of movements in Σ: either right-down or down-right, marked in red and green, respectively in the picture below. These sequences correspond to use the middle red point with coordinates (i + 1 2 , δ i + 1 2 ) or (i + 1 2 , δ i − 1 2 ), respectively. Hence, there are two possibilities for replacing e, unless δ i = δ i+1 = 0. In this case, only the crossed red point (i + 1 2 , 1 2 ) can be used. As a result, if p counts the number of plateaus of Γ with positive height, then there are exactly 2 p different staircase paths with Γ as their skeleton. → Figure 17: A plateau in S(n) (left) and its associated staircase paths in FF(n) (right).
Notice that, since distance-equivalent staircase paths have the same associated skeleton distance path, in particular, they are associated to the same flag distance value, which becomes a numerical invariant that can be assigned to staircase paths. On the other hand, a staircase path always contains n + 1 black dots and n red ones. Using these facts along with Remark 3.14, we obtain the next result. The same idea can be applied to the set of points to the right of Σ, i.e., the points in F(Σ), the Ferrers subdiagram of FF(n) having Σ as their silhouette.  Definition 4.20. Two Ferrers subdiagrams of FF(n) are said to be distanceequivalent if they have the same underlying black diagram. Analogously, two embedded partitions λ and λ ′ are said to be distance-equivalent if their associated Ferrers subdiagrams F λ and F λ ′ are. It is also possible to determine algebraically whether two different subdiagrams in FF(n) are distance-equivalent. To do so, we make use of the corresponding embedded partitions. (1) In case i is even, the number of black dots in the i-th row is λ i 2 for n even, and λ i 2 for n odd.
(2) In case i is odd, the number of black dots in the i-th row is λ i 2 for n even, and λ i 2 for n odd.
Clearly, the number of red points at each row is just λ i minus the corresponding number of black points given above.
if n is even, Notice that the i-th component of the underlying distribution U λ represents the number of black points in the i-th row of the underlying black diagram U F λ . Nevertheless, the underlying distribution U λ of a partition λ is not necessarily a partition itself as we can see in the following example.
In the next result we present a criterion to characterize those distanceequivalent embedded partitions also in terms of their (maybe different) associated underlying distributions. (1) One of these two conditions hold: (2) In any of the cases above, for every 1 i m.
Proof. Assume that λ and λ ′ are distance-equivalent, i.e., their associated Ferrers subdiagrams F λ and F λ ′ are distance-equivalent. Hence, their underlying black diagrams must coincide, that is, U F λ = U F λ ′ . Since one out of two rows in FF(n) starts with a black point, the number of rows of distance-equivalent Ferrers subdiagrams can differ by, at most, one unit. Moreover, the extra row can only contain a red point. In terms of the partitions λ and λ ′ , we conclude that m ′ ∈ {m, m + 1}. Moreover, in case m ′ = m + 1, the first dot (from the right) in the (m + 1)-th row of F λ ′ must be red. That happens if, and only if, m + n is odd and the part λ ′ m ′ = 1. In addition, for the first m rows, the number of black dots in both diagrams have to coincide. Equivalently, for every 1 i m, it must satisfy The converse holds immediately by using the same arguments. At this point we are able to establish in a consistent way the necessary link between embedded partitions in FF(n) and flag codistance values which will permit us to study properties of flag codes in terms of suitable partitions. The next result follows straightforwardly: Due to the previous proposition, we can introduce a new concept that relates embedded partitions and codistance.
Definition 4.26. Given λ = (λ 1 , . . . , λ m ) an embedded partition in FF(n), we say that its underlying distribution U λ splits the value u λ defined in (16), or that it is an splitting of u λ . This value u λ is common for F λ and all its distance-equivalent Ferrers subdiagrams.
Remark 4.27. By extension, if we are in the conditions of Proposition 4.25, we have that u λ =d, and we say that U λ is an splitting of the codistanced. Notice that these splittings are not codistance vectors. Given full flags F, F ′ on F n q , their codistance vector is the sequence of (subspace) codistances between their subspaces. More precisely, the i-th component of this vector is for every 0 i n. Moreover, if d = d f (F, F ′ ), thend is also obtained as the sum of the previous values. However, the notions of splitting and codistance vector (associated tod) represent different ideas.
Finally, the next result provides the bridge to translate the information given by distance paths to the embedded partitions level and conversely. Proof. It suffices to summarize all the results provided along this section. We start from a distance path Γ and consider the flag distance value d = d Γ . This value is an invariant of all the distance-equivalent staircase paths in Σ(Γ) by Corollary 4.17. Take now all the Ferrers subdiagrams FF(n) whose silhouettes are in Σ(Γ), i.e., those ones of the form F Σ for any Σ ∈ Σ(Γ). Each F Σ is also associated to the corresponding partition λ(Σ). Notice that all these Ferrers diagrams are distance-equivalent since they have the set of black points that remains at right of Γ as their common underlying black diagram U , which has exactlyd black points. Hence, all the partitions {λ(Σ) | Σ ∈ Σ(Γ)} are distanceequivalent and their common underlying distribution is, by means of Proposition 4.25, a splitting of the codistanced.
On the other hand, consider a splitting U λ of the codistanced, induced by an embedded partition λ (or any other distance-equivalent partition). The distribution U λ determines the underlying black diagram U F λ of F λ (and of all its distance-equivalent Ferrers subdiagrams). The silhouette Σ = Σ(F λ ) has a skeleton Γ = Γ(Σ) that is a distance path associated to the distance d. This skeleton Γ is common for all the staircase paths that are silhouettes of subdiagrams distance-equivalent to F λ . Hence, every partition providing the distribution U λ leads to the same distance path Γ.

Applications and examples
In this section we show how this new dictionary between flag distance values and underlying distributions of certain partitions can be applied to establish connections between the parameters of a given full flag code and the ones of its projected codes. To this end, we start with a lemma that counts the number of circle black dots of a rectangular Ferrers subdiagram. Proof. Note that, as R has a rows and b columns, it is the Ferrers subdiagram associated to the partition λ = (b, (a) . . ., b) where λ = ab. Then, the number of black points in R is the value u λ (see (16)), given as the sum of the components of the underlying distribution U λ . According to Definition 4.22, the expression of U λ is Hence, for even values of n, we have: In general, note that for every positive integer c, we have c = c 2 + c 2 . Thus, in case that either a or b is even, it holds u λ = ab 2 = ab 2 . On the other hand, if both a and b are odd, then expression (17) becomes and the result is true for n even. If n is odd, the result follows by using the same arguments and taking into account that Next we apply this lemma in order to relate the cardinality of a given flag code to the ones of its projected codes, by counting circle black dots in specific rectangles in the Ferrers diagram frame.
Theorem 5.2. Consider a full flag code C on F n q with codistanced f (C) and take a dimension 1 i n 2 . If the codistance satisfies Proof. Assume that |C| = |C i | and thatd f (C) satisfies (19). In this case, there exist different flags F, F ′ in C with F i = F ′ i . Equivalently, the distance path Γ(F, F ′ ) ∈ Γ(C) passes through the crossed point (i, 0) of S(n), which determines, in turn, a rectangle with i rows and n − i columns over it in the enriched distance supportŜ(n). By means of Lemma 5.1, this rectangle contains exactly if n is odd, circle black dots. Moreorver, notice that, if n is odd, then i(n − i) is even and we can simply write for every value of n. Notice that, at least all these p circle black points remain over the distance path Γ(F, F ′ ), and then they do not contribute to the computation of d f (F, F ′ ). Hence, we have Consequently, we obtaind f (C) = D n − d f (C) p, which is a contradiction, and it must hold |C| = |C i |. The same arguments, but considering a rectangle with n − i rows and i columns, lead to |C| = |C n−i |. On the other hand, if we take a dimension i j n 2 , and we write j = i + k for some integer k 0, then it holds Hence, ifd f (C) satisfies the stated condition, in particular, we also havē and, arguing as above, we get |C| = |C j | = |C n−j |, for every i j n 2 . On the other hand, the connection between distance paths and Ferrers subdiagrams established in the previous section, enables us to associate the following combinatorial objects to a flag code C. In other words, subdiagrams in F(C) are those ones whose silhouettes have a distance path in Γ(C) as their distance skeleton.
Associated to this set of Ferrers subdiagrams, we can consider, in turn, their sets of Durfee rectangles as follows.
Definition 5.4. Let C be a full flag code on F n q . For every 0 k n − 2, the set of Durfee k-rectangles of C is given by In particular, for k = 0, we simply write D(C) = D 0 (C) and speak about the set of Durfee squares of C.
Notice that, given an embedded partition λ = (λ 1 , . . . , λ m ) and an integer 0 k n − 2, the Durfee k-rectangle D k (F λ ) has, at least one row if, and only if, it holds λ 1 k + 1. On the other hand, for those Ferrers diagrams F having no more than k points in their first row, we put D k (F) as the "empty" Durfee k-rectangle, which has zero rows (see Figure 19). Considering this special case makes the set D k (C) contain at least one element, for every 0 k n − 2.
Remark 5.5. Observe that the set D k (C) can be encoded as a set of integers as follows. Assume that where each R k j is a Durfee k-rectangle having 0 r k j n−k 2 rows, thus r k j × (r k j + k) points. Then, to know D k (C), we just need to store the list of integers {r k 1 , . . . , r k m k }. Moreover, without loss of generality, we can assume that r k 1 > r k 2 > · · · > r k m k 0 so that R k 1 is the biggest Durfee k-rectangle in D k (C). As said before, we are also contemplating the possibility of having Durfee krectangles with 0 rows. Hence, in any case, the value r k 1 associated to C always exists.
Let us finally see how can we use Durfee rectangles to relate the parameters of C to the ones of its projected codes.

From the flag distance to subspace distances
By construction, a Durfee k-rectangle in FF(n) having r rows (and then r + k columns) has its left down vertex at the point with coordinates in the enriched distance supportŜ(n). This vertex will help us to obtain information about the projected codes of a given flag code whenever it is a point in the distance support S(n), i.e., if it is a circle black point. Observe also that this happens if, and only if, n−k 2 is an integer or, equivalently, when n and k have the same parity. In this case, k-rectangles give information about the projected code of dimension n−k 2 . The following picture illustrates this fact. Figure 19: Possible Durfee 4-rectangles in FF(8) and 1-rectangles in FF (7), included those ones with zero rows (in dashed lines).
At left, for n = 8, we have three Durfee 4-rectangles with 0, 1 and 2 rows, respectively. Their left down vertices, after rotating back the diagram, are in the column corresponding to the dimension 8−4 2 = 2. At right, different Durfee 1-rectangles on FF(7) are shown. All of them have their left down vertices in the column related to the dimension 7−1 2 = 3. Having this in mind, we start from our full flag code C on F n q and take 0 k n − 2 with the same parity than n. We use the set of k-rectangles D k (C) in order to derive information about the cardinality and minimum distance of the projected code of dimension i = n−k 2 with i ∈ {1, . . . , n 2 }. Conversely, if we are specifically interested in the projected code C i , where 1 i n 2 , then we just need to consider the set of (n − 2i)-rectangles of C.
Theorem 5.6. Let C be a full flag code on F n q and consider a dimension 1 i n 2 . They are equivalent: (i) There are Durfee (n − 2i)-rectangles in D n−2i (C) with r rows, where 0 r i.
Proof. Notice that any (n − 2i)-Durfee rectangle in the Ferrers diagram frame has its lower left vertex in a circle black dot, corresponding to the dimension i in the distance support. Moreover, the number of rows of the rectangle coincides with the number of dots being simultaneously in the rectangle and in S(i, n), i.e., in the i-th column of the distance support S(n) (recall (10) and (11)). Hence, a Durfee (n − 2i)-rectangle in D n−2i (C) with r rows is determined by the existence of Ferrers subdiagrams in F(C) whose silhouettes pass through the black circle point (i, i − r). This happens if, and only if, the skeleton of each of these staircase paths is a distance path in Γ(C) passing through the vertex (i, i − r) as well. Equivalently, there must exist flags F, Now, we focus on the biggest Durfee (n − 2i)-rectangle of C, denoted in Remark 5.5 as R n−2i 1 , and that contains 0 r n−2i 1 i rows. In addition, since we will always work with rectangles in D n−2i (C), we drop the superscripts and simply write R 1 and r 1 , respectively. In light of the previous theorem, we analyze two possible situations concluding the following results.
Theorem 5.7. Assume that C is a full flag code on F n q . Consider a dimension 1 i n 2 and the biggest rectangle R 1 in D n−2i (C). The following statements are equivalent: (1) The number of rows of R 1 , that is r 1 , satisfies 0 r 1 < i.
Proof. By application of Theorem 5.6 and the maximality of r 1 , we know that the existence of Durfee (n − 2i)-rectangles with 0 r 1 < i rows in D n−2i (C) is equivalent to say that, for any choice of different flags F, F ′ ∈ C, we have and the equality holds for some pair of flags in the code. In other words, |C i | = |C| because a couple of flags in C never share their i-th subspaces and, clearly, Now we study the remaining case, in which r 1 attains its maximum possible value, that is, r 1 = i. In that situation, we obtain valuable information about the projected code C i in terms of the second largest rectangle R 2 in D n−2i (C), whenever it exists. More precisely: and assume that the largest rectangle R 1 ∈ D n−2i (C) has r 1 = i rows. Hence, the following statements hold: (1) D n−2i (C) = {R 1 } if, and only if, |C i | = 1 or, equivalently, d I (C i ) = 0.
Proof. By means of Theorem 5.6, the condition r 1 = i is equivalent to say that d I (F i , F ′ i ) = 0 for a pair of different flags F, F ′ ∈ C. This happens if, and only if, Let us distinguish two cases: (1) If R 1 is the only (n − 2i)-rectangle of the code, by Theorem 5.6, we have that d I (F i , F ′ i ) = 0 for every pair of different flags F, F ′ ∈ C. In other words, all the flags in C share their common i-th subspace. Equivalently, the projected code C i consists of a single subspace and d I (C i ) = 0.
(2) On the other hand, if D n−2i (C) = {R 1 }, we can consider the second largest (n − 2i)-rectangle R 2 of C, which has 0 r 2 < r 1 = i rows. Hence, for every pair of different flags F, F ′ ∈ C it holds either d I (F i , F ′ i ) = 0 or d I (F i , F ′ i ) i − r 2 , and the last inequality becomes an equality for some choice of flags in C by means of Theorem 5.6. Equivalently, d I (C i ) = i − r 2 > i − r 1 = 0. Remark 5.9. Observe that Theorems 5.7 and 5.8 give us information about the parameters of all the projected codes of dimensions up to n 2 . For the remaining dimensions, it suffices to reason in the same way by simply considering maximal k-rectangles with c columns and c + k rows. Moreover, it is important to point out that we do not even need to know all the rectangles in D n−2i (C); it suffices to know the largest one and, in the worst case, also the second one.
From the results stated along this subsection, and applying Lemma 5.1, we derive bounds for the minimum distance of the projected codes of C in terms of its codistanced f (C).
Corollary 5.10. Let C be a full flag code on F n q with associated codistancē d f (C). Take a dimension 1 i ⌊ n 2 ⌋ and consider an integer 0 r i. Hence, wheneverd Proof. It suffices to see that, by means of Lemma 5.1, the number of circle black points in a rectangle with r rows and r + n − 2i columns is exactly if n is even, if n is odd.
Morover, notice that, for odd values of n, the product r(r + n − 2i) is always even and then Hence, under condition (20), the number of rows of any Durfee (n−2i)-rectangle in D n−2i (C) is upper bounded by r 1 < r i and we conclude, by means of Theorem 5.7, that |C i | = |C| and d I (C i ) = i − r 1 > i − r.

From subspace distances to the flag distance
In this subsection we deal with the converse problem: given a full flag code on F n q , we obtain information of its parameters from the ones of its projected codes. As before we restrict our study to dimensions 1 i n 2 since, for higher dimensions, it suffices to consider rectangles with more rows than columns.
Theorem 5.11. Let C be a full flag code on F n q and take 1 i (1) If |C i | = |C|, then (2) If |C i | < |C|, then Proof. Let us start assuming that |C i | = |C|. If we write d I (C i ) = i − r for some 0 r i, then Corollary 5.10 leads tō As a consequence, it holds On the other hand, given arbitrary different flags F, F ′ ∈ C, we have that Hence, every distance path in Γ(C) passes either through the vertex (i, d I (C i )) (green vertex in the next figure) or above it. As a result, such a vertex determines a set of circle black dots that always remain under distance paths in Γ(C). This set is exactly the triangle in red in the picture below, which top vertex is the point (i, d I (C i )) and edges with slope 1 (left) and −1 (right). The number of circle black points in such a triangle coincides with the number of circle dots in the distance support S(2d I (C i )), which is d I (C i ) 2 . Hence, Remark 3.14 leads to d f (C) d I (C i ) 2 . Now, if we suppose that |C i | < |C|, the result follows straightforwardly by Theorem 5.2.

Some examples
We finish this section by applying the previous results to three representative situations.
Example 5.12. For n = 8, the full flag distance takes values in the interval [0, 16]. Consider an arbitrary full flag code C with minimum distance d f (C) = 12 or, equivalently, codistanced f (C) = 16−12 = 4. For this value of the codistance, and by application of Theorem 5.2, we can derive that |C| = |C 2 | = · · · = |C 6 |, sinced f (C) = 4 < 2(8−2) 2 = 6. Concerning the projected codes distances, for dimension 4, we look at the number of points in Durfee squares (0-rectangles). Since a Ferrers subdiagram with a Durfee square with 3 rows contains, at least ⌈9/2⌉ = 5 > 4 circle black points, by means of Corollary 5.10, we can ensure that d I (C 4 ) 2. Similarly, for dimensions 2 and 3, we consider 4-rectangles and 2-rectangles, respectively and obtain d I (C 2 ) 1 and d I (C 3 ) 1. These properties are transferred to the projected codes of dimensions 5 and 6 by symmetry of the Ferrers diagram frame. On the other hand, the knowledge of just the code codistance does not give any information about the projected codes C 1 and C 7 . To extract more information in these last cases it is also necessary to know if D 6 (C) contains either one or two elements. There are three possible situations: In the first case, by means of Theorem 5.7, we know that |C 1 | = |C| and d I (C 1 ) = 1. In the remaining cases, we have |C 1 | < |C|. Moreover, Theorem 5.8 leads to d I (C 1 ) = 0 in the second situation and to d I (C 1 ) = 1 in the third one.
Example 5.13. Also for n = 8, now we consider a full flag code C ′ and assume that the projected code C ′ 4 satisfies |C ′ 4 | = |C ′ | and d I (C ′ 4 ) = 2. In this case, by application of Theorem 5.11, we know that 4 d f (C ′ ) 14. Moreover, if, in addition, |C ′ 3 | = |C| and d I (C ′ 3 ) = 1, then Theorem 5.11 leads also to 1 d f (C ′ ) 12 and we conclude that 4 d f (C ′ ) 12.
In general, the more conditions on the projected codes, the more information on the flag code. However, at times we obtain redundant information. For instance, if we require the projected code C ′ 2 to fulfill |C ′ 2 | = |C ′ | and d I (C ′ 2 ) = 1, we obtain 1 d f (C ′ ) 13, which does not provide any new data. Nevertheless, if |C ′ 2 | < |C ′ |, we improve our knowledge about d f (C ′ ) since, by virtue of Theorem 5.11, it must hold 4 d f (C ′ ) 10.
Remark 5.14. As pointed out in Remark 2.13, determining the exact value of d f (C ′ ) from just the list of distances {d I (C i )} 7 i=1 , and conversely, is not always possible. Nevertheless, we have seen that just with the data of at most two Durfee (n − 2i)-rectangles for any dimension i, which is considerably less than knowing the whole set of distance paths of C, we are able to establish interesting connections between flag codes and their projected codes.
Example 5.15. We finish this subsection by looking at the special case of full flag codes with the maximum possible distance. These codes were characterized in [5,Th. 3.11] in terms of their projected codes, which must attain the maximum possible distance for their dimensions and have the same cardinality than the flag code. Here below, we translate that result to the setting introduced in this work to state an alternative combinatorial characterization of optimum distance full flag codes.
Theorem 5.16. Let C be a full flag code on F n q . They are equivalent: (1) d f (C) = D n (ord f (C) = 0).
(2) The set Γ(C) consists of the only distance path passing either through the point ( n 2 , n 2 ), if n is even, or through the points ( n 2 , n 2 ) and ( n 2 , n 2 ), if n is odd.
(3) The set of Ferrers subdiagrams associated to C is Proof. Observe that, by means of Corollary 3.18, the conditiond f (C) = 0 holds if, and only if, distance paths in Γ(C) leave no point above them. In other words, C has a unique distance path: the one that passes trough the points of S(n) having coordinates (i, min{i, n − i}), for all 0 i n (see the picture below). By means of the trident rules (12) and (13), this is exactly the only distance path passing through ( n 2 , n 2 ), if n is even, or through ( n 2 , n 2 ) and ( n 2 , n 2 ), if n is odd. Figure 22: Maximum distance paths with n even (left) and n odd (right).
As a result, statements (1) and (2) are equivalent. Moreover, assuming (2), and according to Definition 5.3, we conclude that only Ferrers diagrams F 0 and F (1) (if n is odd) appear in F(C). On the other hand, if condition (3) holds, the underlying black diagrams associated to the flag code C are empty and then, the associated value of the codistanced f (C) is zero, which finishes the proof.
Remark 5.17. The result given in [5,Th. 3.11] can also be proved in our new combinatorial terms. Observe that condition (3) in the previous theorem is equivalent to say that, for every 1 i ⌊ n 2 ⌋, the set D n−2i (C) contains just one element: the Durfee (n − 2i)-rectangle with zero rows. Hence, by means of Theorem 5.7, we conclude that every projected code of dimension up to ⌊ n 2 ⌋ attains the maximum possible distance, i.e., d I (C i ) = i, and has size |C i | = |C|. For those projected codes of higher dimensions, as stated in Remark 5.9, it suffices to consider rectangles with more columns than rows.
In this paper, we have undertaken a detailed study of the flag distance in terms of different combinatorial objects. To this end, we have first devised a nice way to graphically represent this numerical parameter through distance paths drawn in a distance support whose particular shape has led us to design an associated Ferrers diagram frame. Hence, we have established a one-to-one correspondence between the set of distance paths (associated to a precise distance value d) and the set of underlying distributions of Ferrers subdiagrams having exactlyd (the corresponding codistance value) black points. This fact allows us to perfectly translate properties related to the flag distance into the language of integer partitions. Moreover, we take advantage of this dictionary to suitably associate a family of Ferrers subdiagrams (and their corresponding Durfee rectangles) to a given full flag code. Finally, we show how these objects result very useful to make connections between the parameters of the flag code (minimum distance and cardinality) and the ones of the corresponding projected codes.
In future research, we want apply the dictionary established in this paper to derive new results concerning specific families of full flag codes. Moreover, we would also like to adapt the ideas in the current work to the flag variety of general type vector, where the distance support loses its triangular shape.