lattice and look at the pictures. II. DEFINITIONS, NOTATIONS, THE VERY BASICS The q -binomial coefficient is a natural extension of the standard binomial coefficient The Pochhammer symbol (or shifted factorial) is defined as so that (1)$_{n}$ = n !. Its q -deformed relative, the q -Pochhammer symbol, is defined as The q -binomial coefficient can then be expressed as The q -numbers are defined for any real number a as and it is easy to show that Note that for integers n ≥ 1 we have so that [ n ]$_{q}$ → n when q → 1. To continue, the q -number factorials are defined as 4 [ n ]$_{q}$! = n ∏ k =1 [ k ]$_{q}$ (9) [ n ]$_{q}$ = 1 − q n 1 q = 1 + q + · · · + q $^{n}$− $^{1}$, q = 1 (8) lim $^{q}$→ $_{1}$[ a ]$_{q}$ = a (7) [ a ]$_{q}$ = 1 − q a 1 q , q = 1 (6) [ n k ] q = ( q ; q )$_{n}$ ( q ; q )$_{k}$ ( q ; q )$_{n}$ − k = ( q $^{n}$−k $^{+1}$; q ) k ( q ; q )$_{k}$ (5) ( a ; q )$_{n}$ = $^{n}$− 1 ∏ i =0 ( 1 − a q $^{i}$) , n ≥ 0 (4) ( a )$_{n}$ = $^{n}$− 1 ∏ i =0 ( a + i ) (3) ( n k ) = n ! k ! ( n − k )! , 0 ≤ k ≤ n (2) [ n k ] q = k ∏ i =1 1 − q $^{n}$−i +1 1 − q i , q = 1, 0 ≤ k ≤ n (1)