The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the γ-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions p −k (n) for all k ≥ 2. This enables us to extend the k-colored partition function multiplicatively to a function on k-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.

As shown by Bousquet-Mélou-Claesson- Dukes-Kitaev (2010), ascent sequences can be used to encode (2 + 2)-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power seriesIn this paper, we present a novel way to recursively decompose ascent sequences, which leads to:• a calculation of the Euler-Stirling distribution on ascent sequences, including the numbers of ascents (asc), repeated entries (rep), zeros (zero) and maximal entries (max).In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of zero and max. • a far-reaching generalization of the generating function formula for (asc, zero) due toJelínek. This is accomplished via a bijective proof of the quadruple equidistribution of (asc, rep, zero, max) and (rep, asc, rmin, zero), where rmin denotes the right-to-left minima statistic of ascent sequences. • an extension of a conjecture posed by Levande, which asserts that the pair (asc, zero) on ascent sequences has the same distribution as the pair (rep, max) on (2 − 1)-avoiding inversion sequences. This is achieved via a decomposition of (2 − 1)-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (asc, rep, zero, max) and (rep, asc, max, zero) are equidistributed on ascent sequences.

Restricted growth functions (RGFs) avoiding the pattern $1212$ are in natural bijection with noncrossing partitions. Motivated by recent work of Campbell et al., we study five classical statistics bk, ls, lb, rs and rb on $1212$-avoiding RGFs. We show the equidistribution of (ls, rb, lb, bk) and (rb, ls, lb, bk) on $1212$-avoiding RGFs by constructing a simple involution. To our surprise, this result was already proved by Simion 22 years ago via an involution on noncrossing partitions. Our involution, though turns out essentially the same as Simion's, is defined quite differently and has the advantage that makes the discussion more transparent. Consequently, a multiset-valued extension of Simion's result is discovered. Furthermore, similar approach enables us to prove the equidistribution of (mak, rb, rs, bk) and (rb, mak, rs, bk) on $1212$-avoiding RGFs, where "mak" is a set partition statistic introduced by Steingrímsson.Through two bijections to Motzkin paths, we also prove that the triple of classical permutation statistics (exc+1, den, inv — exc) on $321$-avoiding permutations is equidistributed with the triple (bk, rb, rs) on $1212$-avoiding RGFs, which generalizes another result of Simion. In the course, an interesting $q$-analog of the $\gamma$-positivity of Narayana polynomials is found.

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on R(w), the rearrangement class of a given word w. We construct such an involution in this paper, which yields a comparable joint equidistribution between two sextuple statistics over R(w). Our involution builds on Burstein's involution and Foata-Schützenberger's involution that utilizes the celebrated RSK algorithm.Date: October 22, 2018.