Related papers: Eulerian quasisymmetric functions
This paper is devoted to the family $\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients $(\ell_1+...+\ell_n)!/(\ell_1!...\ell_n!)$, where $n\in\ZZ_{\ge 1}$.…
We present the singular Euler--Maclaurin expansion, a new method for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. In contrast to the traditional…
Let the symmetric functions be defined for the pair of integers $\left( n,r\right) $, $n\geq r\geq 1$, by $p_{n}^{\left( r\right) }=\sum m_{\lambda }$ where $m_{\lambda }$ are the monomial symmetric functions, the sum being over the…
Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur…
The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian…
In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra $U_q \big( \widehat{\mathfrak{sl}} (1 | n)…
In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent…
We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of…
We introduce the theory of normal ordered grammars, which gives a natural generalization of the normal ordering problem. To illustrate the main idea, we explore normal ordered grammars associated with the Eulerian polynomials and the…
It is widely accepted nowadays that polyzetas are connected by polynomial relations. One way to obtain relations among polyzetas is to consider their generating series and the relations among these generating series. This leads to the…
In the 1995 paper entitled "Noncommutative symmetric functions," Gelfand, et. al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions, along with analogues for the elementary and the…
The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial…
Linear harmonic number sums had been studied by a variety of authors during the last centuries, but only few results are known about nonlinear Euler sums of quadratic or even higher degree. The first systematic study on nonlinear Euler sums…
We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent…
The quasisymmetic Macdonald polynomials $G_{\gamma}(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_{\lambda}(X; q, t)$ with the…
We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun…
Recently there has been a renewed interest in asymptotic Euler-MacLaurin formulas, partly due to applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth…
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…
The main purpose of this paper is to present a systemic study of some families of multiple $q$-Euler numbers and polynomials. In particular, by using the $q$-Volkenborn integration on $\Bbb Z_p$, we construct $p$-adic $q$-Euler numbers and…