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Richard P. Stanley

Publications and source records attributed to Richard P. Stanley.

At least 19 recordsLinked to original sources
arXiv2026

Sprout Symmetric Functions: Part 1

A \emph{sprout sequence} is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq 0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is called the \emph{seed} of $\frakr$. This concept originated in the work of Littlewood and Richardson (though not with the name ``sprout sequence''), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case $F(t)=\sec(\sqrt{t})$. We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of $R_n$ for all $n$ are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The expansion of $R_n$ in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for $R_n$ as a sum of chromatic symmetric functions of interval orders.

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arXiv2024

A Shifted Parking Function Symmetric Function

We define a "shifted analogue" $\mathrm{SH}_n$ of the parking function symmetric function $\mathrm{PF}_n$. The expansion of $\mathrm{SH}_n$ in terms of three bases for shifted symmetric functions is explicitly described. We don't know a shifted analogue for parking functions themselves, but some desirable properties of such an analogue are discussed.

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arXiv2023

The Redei--Berge symmetric function of a directed graph

Let $D=\left( V,A\right) $ be a digraph with $n$ vertices, where each arc $a\in A$ is a pair $\left( u,v\right) $ of two vertices. We study the \emph{Redei--Berge symmetric function} $U_{D}$, defined as the quasisymmetric function% \[ \sum L_{\operatorname*{Des}\left( w,D\right) ,\ n}\in\operatorname*{QSym}. \] Here, the sum ranges over all lists $w=\left( w_{1},w_{2},\ldots ,w_{n}\right) $ that contain each vertex of $D$ exactly once, and the corresponding addend is% \[ L_{\operatorname*{Des}\left( w,D\right) ,\ n}:=\sum_{\substack{i_{1}\leq i_{2}\leq\cdots\leq i_{n};\ı_{p}<i_{p+1}\text{ for each }p\text{ satisfying }\left( w_{p},w_{p+1}\right) \in A}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}% \] (an instance of Gessel's fundamental quasisymmetric functions). While $U_{D}$ is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express $U_{D}$ in terms of the power-sum symmetric functions. We show that $U_{D}$ is always $p$-integral, and furthermore is $p$-positive whenever $D$ has no $2$-cycles. When $D$ is a tournament, $U_{D}$ can be written as a polynomial in $p_{1},2p_{3},2p_{5},2p_{7},\ldots$ with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-$4$ refinement of Redei's theorem.

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arXiv2023

Some enumerative properties of parking functions

A parking function is a sequence $(a_1,\dots, a_n)$ of positive integers such that if $b_1\leq\cdots\leq b_n$ is the increasing rearrangement of $a_1,\dots,a_n$, then $b_i\leq i$ for $1\leq i\leq n$. In this paper we obtain some new results on the enumeration of parking functions. We will consider the joint distribution of several sets of statistics on parking functions. The distribution of most of these individual statistics is known, but the joint distributions are new. Parking functions of length $n$ are in bijection with labelled forests on the vertex set $[n]=\{1,2,\dots,n\}$ (or rooted trees on $[n]_0=\{0,1,\dots,n\}$ with root $0$), so our results can also be applied to labelled forests. Extensions of our techniques are discussed.

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arXiv2023

Posets of width two and skew Young diagrams

Let $P$ be a finite poset of width two, i.e., with no three-element antichain. We associate with $P$ a skew Young diagram $Υ(P)$ and discuss some of the properties of the map $Υ$. In particular, if we regard $Υ(P)$ as a poset in a standard way, then the linear extensions of $P$ are in bijection with the order ideals of $Υ(P)$.

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arXiv2021

Theorems and Conjectures on Some Rational Generating Functions

Let $I_n(x)=\prod_{i=1}^n \left( 1+x^{F_{i+1}}\right)$, where $F_{i+1}$ denotes a Fibonacci number. Let $v_r(n)$ denote the sum of the $r$th powers of the coefficients of $I_n(x)$. Our prototypical result is that $\sum_{n\geq 0} v_2(n)x^n= (1-2x^2)/(1-2x-2x^2+2x^3)$. We give many related results and conjectures. A certain infinite poset $\mathfrak{F}$ is naturally associated with $I_n(x)$. We discuss some combinatorial properties of $\mathfrak{F}$ and a natural generalization, including a symmetric function that encodes the flag $h$-vector of $\mathfrak{F}$.

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arXiv2020

On a generalization of Lie($k$): a CataLAnKe theorem

We initiate a study of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group $S_{2n-1}$ on the multilinear component of the free LAnKe with $2n-1$ generators is given by an irreducible representation whose dimension is the $n$th Catalan number. This leads to a more general result on eigenspaces of a certain linear operator, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape as a consequence of our central result.

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arXiv2019

A note on the asymptotics of the number of O-sequences of given length

We look at the number $L(n)$ of $O$-sequences of length $n$. Recall that an $O$-sequence can be defined algebraically as the Hilbert function of a standard graded $k$-algebra, or combinatorially as the $f$-vector of a multicomplex. The sequence $L(n)$ was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants $c_1$ and $c_2$ and all $n>2$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.

math.AC↗
arXiv2019

Some Linear Recurrences Motivated by Stern's Diatomic Array

We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The proof technique yields a vast generalization. In certain cases we can be more explicit about the resulting linear recurrence.

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arXiv2019

Proof of the Gorenstein Interval Conjecture in low socle degree

Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever $(1,h_1,\dots,h_i,\dots,h_{e-i},\dots,h_{e-1},1)$ and $(1,h_1,\dots,h_i+α,\dots,h_{e-i}+α,\dots,h_{e-1},1)$ are both Gorenstein Hilbert functions for some $α\geq 2$, then $(1,h_1,\dots,h_i+β,\dots,h_{e-i}+β,\dots,h_{e-1},1)$ is also Gorenstein, for all $β=1,2,\dots,α-1$. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case $e\le 5$, in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry.

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arXiv2018

A generalization of a 1998 unimodality conjecture of Reiner and Stanton

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $k\ge 5$, the polynomials $$f(k,m,b)(q)=\binom{m}{k}_q-q^{\frac{k(m-b)}{2}+b-2k+2}\cdot\binom{b}{k-2}_q$$ are nonnegative and unimodal for all $m\gg_k 0$ and $b\le \frac{km-4k+4}{k-2}$ such that $kb\equiv km$ (mod 2), with the only exception of $b=\frac{km-4k+2}{k-2}$ when this is an integer. Using the KOH theorem, we combinatorially show the case $k=5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m,b)$ for $k\le 5$. (This also provides an isolated counterexample to Reiner-Stanton's conjecture when $k=3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.

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arXiv2017

Some Schubert shenanigans

We give a conjectured evaluation of the determinant of a certain matrix $\tilde{D}(n,k)$. The entries of $\tilde{D}(n,k)$ are either 0 or specializations $\mathfrak{S}_w(1,\dots,1)$ of Schubert polynomials. The conjecture implies that the weak order of the symmetric group $S_n$ has the strong Sperner property. A number of peripheral results and problems are also discussed.

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arXiv2016

Smith Normal Form in Combinatorics

This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.

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arXiv2015

Some asymptotic results on q-binomial coefficients

We look at the asymptotic behavior of the coefficients of the $q$-binomial coefficients (or Gaussian polynomials) $\binom{a+k}{k}_q$, when $k$ is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.

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arXiv2015

The Smith Normal Form Distribution of a Random Integer Matrix

We show that the density $μ$ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities $μ_{p^s}$ of SNF over $\mathbb{Z}/p^s\mathbb{Z}$ with $p$ a prime and $s$ some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for $μ_{p^s}$ and compute the density $μ$ for several interesting types of sets. Finally, we determine the maximum and minimum of $μ_{p^s}$ and establish its monotonicity properties and limiting behaviors.

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arXiv2015

The Smith Normal Form of a Specialized Jacobi-Trudi Matrix

Let $\mathrm{JT}_λ$ be the Jacobi-Trudi matrix corresponding to the partition $λ$, so $\det\mathrm{JT}_λ$ is the Schur function $s_λ$ in the variables $x_1,x_2,\dots$. Set $x_1=\cdots=x_n=1$ and all other $x_i=0$. Then the entries of $\mathrm{JT}_λ$ become polynomials in $n$ of the form ${n+j-1\choose j}$. We determine the Smith normal form over the ring $\mathbb{Q}[n]$ of this specialization of $\mathrm{JT}_λ$. The proof carries over to the specialization $x_i=q^{i-1}$ for $1\leq i\leq n$ and $x_i=0$ for $i>n$, where we set $q^n=y$ and work over the ring $\mathbb{Q}(q)[y]$.

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arXiv2015

Unimodality of partitions with distinct parts inside Ferrers shapes

We investigate the rank-generating function $F_λ$ of the poset of partitions contained inside a given shifted Ferrers shape $λ$. When $λ$ has four parts, we show that $F_λ$ is unimodal when $λ=\langle n,n-1,n-2,n-3 \rangle$, for any $n\ge 4$, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form $λ=\langle n,n-t,n-2t,n-3t \rangle$, for any given $t\ge 2$ and $n$ large enough with respect to $t$. When $λ$ has $b\le 3$ parts, we show that our rank-generating functions $F_λ$ are all unimodal. However, the situation remains mostly obscure for $b\ge 5$. In general, the type of results that we obtain present some remarkable similarities with those of the 1990 paper of D. Stanton, who considered the case of partitions inside ordinary (straight) Ferrers shapes. Along the way, we also determine some interesting $q$-analogs of the binomial coefficients, which in certain instances we conjecture to be unimodal. We state several other conjectures throughout this note, in the hopes to stimulate further work in this area. In particular, one of these will attempt to place into a much broader context the unimodality of the posets $M(n)$ of staircase partitions, for which determining a combinatorial proof remains an outstanding open problem.

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