Abstract:

We introduce a weight-dependent extension of the inversion statistic, a classical Mahonian statistic on permutations. This immediately gives us a new weightdependent extension of n!. When we restrict to 312-avoiding permutations, our extension gives rise to a weight-dependent family of Catalan numbers, which happen to coincide with the weighted Catalan numbers that were previously introduced by Postnikov and Sagan by weighted enumeration of Dyck paths [4]. While Postnikov and Sagan’s main focus was on the modulo 2 divisibility of the weighted Catalan numbers, we worked out further properties of these numbers that extend those of the classical case, such as their recurrence relation, their continued fraction, and Hankel determinants. We also discovered an intriguing closed form evaluation of the weighted Catalan numbers for a specific choice of weights. We further present bi-weighted Catalan numbers that generalize Garsia and Haiman’s q; t-Catalan numbers [2], and again satisfy remarkable properties. These are obtained by refining the weighted Catalan numbers by introducing an additional statistic, namely a weightdependent extension of Haglund’s bounce statistic [1, 3].

This is joint work with Shishuo Fu (Chongqing University).

[1] A. Garsia and J.. Haglund, A proof of the q; t-Catalan positivity conjecture, Discrete Math. 256(3) (2002) 677–717.

[2] A. Garsia and M. Haiman, A remarkable q; t-Catalan sequence and q-Lagrange inversion, J. Alg. Combin. 5(3) (1996) 191–244.

[3] J. Haglund, Conjectured statistics for the q; t-Catalan numbers, Adv. Math. 175(2) (2003) 319–334.

[4] A. Postnikov and B.E. Sagan, What power of two divides a weighted Catalan number?, J. Combin. Theory Ser. A 114(5) (2007) 970–977.