Fix-mahonian calculus III; a quadruple distribution

Abstract

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where "des" and "exc" stand for the number of descents and excedances, respectively.

The $q$-tangent and $q$-secant numbers via basic Eulerian polynomials

“…We have p = 1 3 4 14, so that pix σ = 4, inv(12 2 5 11 15) = 3, inv(8 6 7) = 2, inv(13 9 10) = 2, so that lec σ = 7. The next theorem, already derived in [10], is given a new direct proof. The identity…”

“…The analytic proof, given in Section 2, is based on a recent result due to Shareshian and Wachs [19] who made an explicit study of the statistic "maj − exc". The combinatorial proof, given in Section 4, is based on the geometry of alternating permutations, as introduced by André [1], [2] and on an equidistribution property between two three-variable statistics (exc, fix, maj) and (lec, pix, inv) established in [10]. However, to make this paper self-contained, we give a new proof of that equidistribution property by directly calculating the distribution of the latter statistic (see Theorem 4, Section 3).…”

“…To derive the hook factorization of a permutation, it suffices to start from the right and at each step determine the right factor, which is a hook. Those two statistics have been introduced and used in [10]. Furthermore, let Desar n be the subset of S n of the permutations σ, called desarrangements, having the property that pix σ = 0 [7].…”

Eulerian Polynomials: From Euler’s Time to the Present

“…A further step can be made by calculating the exponential generating function for the polynomials A n (s, t, q) := σ∈S n s exc σ t des σ q maj σ (n ≥ 0), as was done in [FH08]: n≥0 A n (s, t, q) u n (t; q) n+1 = r≥0 t r (1 − sq)(usq; q) r ((u; q) r − sq(usq; q) r )(1 − uq r ) .…”

Signed words and permutations, V; a sextuple distribution

“…In [FoHa07a] and [FoHa07b] we have respectively calculated the generating functions for the polynomials B n (1, t, q, Y 0 , Y 1 , Z) and A n (s, t, q, Y 0 ) , where and shown that by giving certain variables specific values the former statistical results on S n and B n could be reobtained. Note that, using the previous definitions of "fexc," "fdes" and "fmaj," the latter polynomial is nothing but B n (s 1/2 , t 1/2 , q 1/2 , Y, 0, 0).…”