Extending Fine sequences: a link with forbidden patterns
  We propose a natural, bivariate, generalization of the nonsingular similarity relations considered by T. Fine. We also provide an enumeration formulae and a generating tree for those relations. The latter allow us to give a new bijection between 321-avoiding derangements and Fine sequences. Moreover, we establish that two special cases are in a one-to-one correspondence with subsets of permutations characterized by forbidden subsequences on the symmetrical group. All our results are established using the technique of generating tree, thus giving entirely bijective proofs. 
Olivier Guibert and Sylvain Pelat-Alloin