Statistics of partial permutations via Catalan matrices

A generalized Catalan matrix (a_n,k)_n,k≥0 is generated by two seed sequences s=(s₀,s₁,…) and t=(t₁,t₂,…) together with a recurrence relation. By taking s_ℓ=2ℓ+1 and t_ℓ=ℓ² we can interpret a_n,k as the number of partial permutations, which are n×n 0,1-matrices of k zero rows with at most one 1 in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on two interesting permutation families, namely the connected permutations and cycle-up-down permutations, are also given.