Related papers: Decorated square paths at q=-1

In [The Delta Conjecture, Trans. Amer. Math. Soc., to appear] Haglund, Remmel, Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This…

In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the…

Let q be a power of a prime and E be an elliptic curve defined over F_q. In "Combinatorial aspects of elliptic curves" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the…

We determine all positive harmonic functions for a large class of "semi-isotropic" random walks on the lamplighter group, i.e., the wreath product of the cyclic group of order q with the infinite cyclic group. This is possible via the…

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by…

A new representation of Dirac's delta-distribution, based on the so-called q-exponentials, has been recently conjectured. We prove here that this conjecture is indeed valid.

Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor…

In this article we prove the $e$-positivity of $G_{\mathbf{\nu}}[X;q+1]$ when $G_{\mathbf{\nu}}[X;q]$ is a vertical strip LLT polynomial. This property has been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and…

N.Wallach has considered an element of the group algebra of the symmetric group S_n which is the sum of an n-cycle, an (n-1)-cycle,...,a 2-cycle and the identity. He showed that multiplication by this element has eigenvalues…

We prove that $\omega \Delta'_{e_{k}}e_n|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all…

We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and polynomials. The first states that the cycle type Eulerian quasisymmetric function $Q_{\lambda,j}$ is Schur-positive, and moreover that the…

We address the problem of enumerating paths in square lattices, where allowed steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We consider two such lattices differing in whether the (1,1) steps are allowed along…

The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant…

For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let…

I continue the investigation of a q-analogue of the convolution on the line started in a joint work with Koornwinder and based on a formal definition due to Kempf and Majid. Two different ways of approximating functions by means of the…

In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group $\mathcal{S}_{N}$. The…

In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated…

The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(\alpha,\beta)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian…

Consider the set of all error--correcting block codes over a fixed alphabet with $q$ letters. It determines a recursively enumerable set of points in the unit square with coordinates $(R,\delta)$:= {\it (relative transmission rate, relative…

For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ (where $V_q$ is the $n$-dimensional vector representation) satisfies Schur--Weyl duality with respect to the commuting actions of the quantized enveloping algebra…