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CMFS-actions on (6512)(8347)
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A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting p...
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... and Wang [26] defined the group action of Z n 2 on D n via the involutions τ c S over all S ⊆ [n]; this group action is called the cyclic modified Foata-Strehl action, abbreviated CMFS-action, see Figure 5 for an illustration. For any permutation σ ∈ S n , let Orb(σ) = {g(σ) : g ∈ Z n 2 } be the orbit of σ under the CMFS-action. ...
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Citations
... Such connections are valuable as they bear combinatorial information which naturally lead to refinements and generalizations of the enumerative results for the combinatorial objects involved. See for example [11,12,17,19,21,22,30,31,34], especially the two surveys by Corteel-Kim-Stanton [13], and Zeng [35] and the references therein. ...
... Comparing to the classic mappings Φ FV and Φ FZ , the mapping Ψ YZL [34], as well as its variant Φ YZL introduced in this paper are quite recent. The work of Han-Mao-Zeng [21] indicates that Ψ YZL actually fits well with Ψ FV by placing Ψ YZL in the first triangle in Fig. 6, where the mapping Ψ SZ was first constructed by Shin-Zeng [30]. Viewing these two factorizations in Fig. 6, one wonders if a similar factorization could be carried out for Φ FZ and Φ YZL . ...
Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Fran\ccon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement.
... It is well known [8,19,25] that the statistics``des"" and``exc"" are equidistributed over permutations of [n] := \{ 1, . . . , n\} , their common generating function being the Eulerian polynomials A n (t), i.e., Since MacMahon's pioneering work [17], various combinatorial variants and refinements of Eulerian polynomials have appeared; see [2,12,13,16,18,24] for some recent papers. ...
... In this paper we shall take a different approach to their problems through the combinatorial theory of J-continued fractions developed by Flajolet and Viennot in the 1980s [7,11]; see [2,6,13,24] for recent developments of this theory. Recall that a J-type continued fraction is a formal power series defined by \infty \sum n=0 a n z n = 1 1 ...
... 1 -wz -t\lambda y z 2 1 -(w + t + 1)z -t(\lambda + 1)(y + 1) z 2 \cdot \cdot \cdot (1.2) with \gamma n = w + n(t + 1) and \beta n = t(\lambda + n -1)(y + n -1). It is known that A n (t, 1, 1, 1) equals the Eulerian polynomial A n (t); see [13,24]. Recently Sokal and the third author [24] have generalized the J-fraction for Eulerian polynomials in infinitely many indeterminates, which are also generalizations of the polynomials A n (t, \lambda , y, w). ...
... Various combinatorial variants and refinements of Eulerain polynomials have recently appeared in [2,6,10,13,14,19]. For a permutation σ := σ(1)σ(2) · · · σ(n) of 1 . . . ...
... Moreover we show that (pex, cyc) is equidistributed with many other bistatistics (see Theorem 1.6). Our main tool is the combinatorial theory of J-continued fractions developed by Flajolet and Viennot in the 1980's [8,9], see [2,5,10,19] for recent developments of this theory. Recall that a J-type continued fraction is a formal power series defined by ∞ n=0 a n z n = 1 ...
... with γ n = w + n(t + 1) and β n = t(λ + n − 1)(y + n − 1). The polynomials A n (t, 1, 1, 1) are the Eulerian polynomials A n (t), see [7,10,19]. As we will show later, the generating functions of the aforementioned statistics are all specialisations of A n (t, λ, y, w). ...
In a recent paper ({arXiv:2101.01928v1}) Baril and Kirgizov posed two conjectures on the equidistibution of and , where cyc, des and exc are classical statistics counting the numbers of cycles, descents and excedances of permutation, while and pex are numbers of special descents and excedances over permutations. In this paper, using combinatorial theory of J-continued fractions and bijections we confirm and strengthen the second conjecture and expand the first one.
... Recently, many different refinements and generalizations of Propositions 1, 2 and 3 have been studied, see [22,25,28,38,42,43] and references therein. In the next section, we first present the main results of this paper and then we give some applications. ...
... The reader is referred to [22,27,32,38] for some recent results related to the joint distribution of excedances, fixed points and cycles. From Proposition 2, we see that if q > 0 is a given real number, then A n (x, 0, q) are γ-positive for n 1. Comparing (2) with (4), we get 2 n A n (x, 1/2, 1) = d B n (x). ...
... By using (22), it is routine to derive the following recurrence system: 1 (x) = 0. Let (a n (x, q), b n (x, q)) be the symmetric decomposition of A n (x, q). It follows from (23) that a (k) n (x) = k n a n (x, 1/k), b (k) n (x) = k n b n (x, 1/k). ...
The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or not, colored or not. Let and be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-gamma-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly increasing, and so they are unimodal with modes in the middle, where fix and cyc are the fixed point and cycle statistics. When p=1 and q=1/2, we find a combinatorial interpretation of the bi-gamma-coefficients of the (p,q)-Eulerian polynomials. We then study excedance and flag excedance statistics of signed permutations and colored permutations. In particular, we establish the relationships between the (p,q)-Eulerian polynomials and some multivariate Eulerian polynomials. Our results unify and generalize a variety of recent results.
... 2. Main results. We follow [4,17,16,22,23] for notations and the nomenclature of various permutation statistics. First we recall three classical involutions defined on S n , namely, the reverse, complement and the composition of the two. ...
... Recently, Yan-Zhou-Lin [29] constructed a bijection ψ Y ZL from S n+1 to LH n . Han-Mao-Zeng [16] showed that Yan-Zhou-Lin's bijection ψ Y ZL is a composition of Françon-Viennot's bijection and Shin-Zeng's bijection Ψ, see [ ...
... This completes the proof of (16). This lead to (17) combining (39) and (16). ...
Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.
We introduce a kind of -Catalan numbers of Type A by generalizing the Jacobian type continued fraction formula, we proved that the corresponding expansions could be expressed by the polynomials counting permutations on by various descent statistics. Moreover, we introduce a kind of -Catalan numbers of Type B by generalizing the Jacobian type continued fraction formula, we proved that the Taylor coefficients and their -coefficients could be expressed by the polynomials counting permutations on by various descent statistics. Our methods include permutation enumeration techniques involving variations of bijections from permutation patterns to labeled Motzkin paths and modified Foata-Strehl action.
The binomial Eulerian polynomials, first introduced in work of Postnikov, Reiner and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations and proved that they can be written as the sums of two γ-positive polynomials. In this paper, we find combinatorial interpretations of Athanasiadis' γ-positive polynomials, which leads to an alternative proof of their γ-positivity expansions using the method of group actions on colored permutations. Two results are presented. The first one is to give the γ-coefficients of the symmetric decompositions of binomial Eulerian polynomials for colored permutations, which answers a problem of Athanasiadis (2020) [3]. The second one is to give a direct combinatorial proof on the γ-expansions in the symmetric decompositions of colored derangement polynomials, which answers another problem asked by Athanasiadis (2018) [2].
In 2008 Brändén proved a (p,q)-analogue of the γ-expansion formula for Eulerian polynomials and conjectured the divisibility of the γ-coefficient γn,k(p,q) by (p+q)k. As a follow-up, in 2012 Shin and Zeng showed that the fraction γn,k(p,q)/(p+q)k is a polynomial in N[p,q]. The aim of this paper is to give a combinatorial interpretation of the latter polynomial in terms of André permutations, a class of objects first defined and studied by Foata, Schützenberger and Strehl in the 1970s. It turns out that our result provides an answer to a recent open problem of Han, which was the impetus of this paper.
