On the Parity of Certain Coefficients for a q-Analogue of the Catalan Numbers

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, Cn, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan. In particular, it is well known that Cn is odd if and only if n = 2 k − 1 for some k > 0. The polynomial F ch n (321; q) = ∑ σ∈Avn(321) q ch(σ), where Avn(321) is the set of permutations in Sn that avoid 321 and ch is the charge statistic, is a q-analogue of the Catalan numbers since specializing q = 1 gives Cn. We prove that the coefficient of q i in F ch 2k−1(321; q) is even if i > 1, giving a refinement of the “if” direction of the Cn parity result. Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index. In addition, Sagan and Savage have recently defined a notion of st-Wilf equivalence for any permutation statistic st and any two sets of permutations Π and Π′. We say Π and Π′ are st-Wilf equivalent if ∑ σ∈Avn(Π) q st(σ) = ∑ σ∈Avn(Π′) q st(σ). In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of S3.