The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π)$ is the alternating major index of $π$. This leads us to discover a $q$-analog of $n!=2^{\ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $γ$-vectors of the alternating descent polynomials by using these two recursions and the ${\textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
We examine an elementary problem on prime divisibility of binomial coefficients. Our problem is motivated by several related questions on alternating groups.
Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=cs~{G}$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(cs~{G})$ by $D(G)$. In this paper we will find the number of connected components of $D(G)$ where $G$ is the symmetric group $S_n$ or is the alternating group $A_n$.
Gamma-positivity appears frequently in finite geometries, combinatorics and number theory. Motivated by the recent work of Sagan and Tirrell (Adv. Math., 374 (2020), 107387), we study the relationships between gamma-positivity and alternating gamma-positivity. As applications, we derive several alternatingly gamma-positive polynomials related to Narayana polynomials and Eulerian polynomials. In particular, we show the alternating gamma-positivity and Hurwitz stability of a combination of the modified Narayana polynomials of types A and B. By using colored $2\times n$ Young diagrams, we present…
In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.
We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric function and many other invariants have a property we call positively $h$-alternating. This property of positively $h$-alternating leads to Schur positivity and $e$-positivity when applying the operator $\nabla$ at $q=1$. We conclude by showing that the invariants we consider can be expressed as scheduling problems.
We say that a permutation $π=π_1π_2\cdots π_n \in \mathfrak{S}_n$ has a peak at index $i$ if $π_{i-1} < π_i > π_{i+1}$. Let $\mathcal{P}(π)$ denote the set of indices where $π$ has a peak. Given a set $S$ of positive integers, we define $\mathcal{P}_S(n)=\{π\in\mathfrak{S}_n:\mathcal{P}(π)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| \mathcal{P}_S(n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the …
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 $p\in [0,1]$ and $q\in [0,1]$ 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 incre…
The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $A_n^+(t)$ as the polynomial obtained when we sum descents over the alternating group. We show that $A_n^+(t)$ is gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2$ (mod 4) we show that $A_n^+(t)$ can be written as a sum of two gamma positive polynomials while if $n \equiv 3$ (mod 4), we show that $A_n^+(t)$ can be written as a sum of three gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials.
The aim of this paper is to make a systematical study on the stability of polynomials in combinatorics. Applying the characterizations of Borcea and Brändén concerning linear operators preserving stability, we present criteria for real stability and Hurwitz stability. We also give a criterion for Hurwitz stability of the Turán expressions. As applications, we derive some stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating runs polynomials of types $A$ and $B$ and solve a conjecture concerning Hurwit…
The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that $\mathsf{AExc^{+}}_n(t)$ is gamma positive iff $n \geq 5$ and $n \equiv 1$ (mod 2). When $n \geq 4$, and $n \equiv 0$ (mod 2) we show that $\mathsf{AExc^{+}}_n(t)$ can be written as a sum of two gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials. Finally, we show gamma positivity results when we…
Let $1\leq k \leq n$ and let $X_n = (x_1, \dots, x_n)$ be a list of $n$ variables. The {\em Boolean product polynomial} $B_{n,k}(X_n)$ is the product of the linear forms $\sum_{i \in S} x_i$ where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots, n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call {\em Chern plethysm}. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a com…
Motivated by the work of Visontai and Dey-Sivasubramanian on the gamma-positivity of some polynomials, we find the commutative property of a pair of Eulerian operators. As an application, we show the bi-gamma-positivity of the descent polynomials on permutations of the multiset $\{1^{a_1},2^{a_2},\ldots,n^{a_n}\}$, where $0\leqslant a_i\leqslant 2$. Therefore, these descent polynomials are all alternatingly increasing, and so they are unimodal with modes in the middle.
In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$-$q$-log-convexity of alternating Eulerian polynomials, $2$-log-convexity of Eu…
