Permutations with a Given X-Descent Set

TL;DR

This work introduces and analyzes the X-descent set statistic for permutations, defining $d_X(I;n)$ and its related class $\mathcal{D}_X(I;n)$. It develops a general subset-sum recurrence for $d_X(I;S)$ and shows that, under standardization-invariance, this reduces to a binomial-type recursion that expresses $d_X(I;n)$ in terms of $d_X(I^-;n)$ and $d_X(\emptyset;n)$. A key insight is that many quantities are governed by the simpler $d_X(\emptyset;n)$, which counts Hamiltonian paths in an associated digraph $G_n(X)$; the paper then provides methods to compute or bound these path counts via tournament formulas, periodic-transfer matrices, and inclusion-exclusion in special X-classes. It also proves a probabilistic lower bound demonstrating that $d_X(\emptyset;n)$ typically grows faster than any fixed polynomial in $n$, and closes with open problems and directions for further exploration.

Abstract

Building on the work of Grinberg and Stanley, we begin a systematic study of permutations with a prescribed $X$-descent set. In particular, for a set $X \subseteq \mathbb{N}^2$, and $I \subseteq [n-1]$, we study the permutations $π\in \mathfrak{S}_n$ whose $X$-descent set is precisely $I$, meaning $(π_i,π_{i+1}) \in X$ precisely when $i \in I$. The central focus is enumerating these permutations for a fixed $X,I$ and $n$: this count is denoted by $d_X(I;n)$. We derive a recursion which under expected conditions simplifies to a binomial-type recurrence determined entirely by the values $d_X(\emptyset;n)$. This extends the work of Díaz-Lopez et al.\ on descent polynomials. The resulting reduction shows that the general statistic $d_X(I;n)$ is typically governed by the ``descent-free'' quantities $d_X(\emptyset;n)$, motivating a closer analysis of these numbers. We observe that $d_X(\emptyset;n)$ enumerates Hamiltonian paths in a directed graph canonically associated to $X$. We then record several families of sets $X$ for which $d_X(\emptyset;n)$ is explicit or effectively computable. This includes families with periodicity for which transfer matrix methods apply, and families with succession-type relations where inclusion-exclusion applies. We then investigate the typical behavior of $d_X(\emptyset;n)$ from a probabilistic perspective.

Theorems & Definitions (32)

• Definition 1.1
• Definition 1.2
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• proof