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Jeffrey B. Remmel

Publications and source records attributed to Jeffrey B. Remmel.

At least 19 recordsLinked to original sources
arXiv2022

Positional Marked Patterns in Permutations

We define and study positional marked patterns, permutations $τ$ where one of elements in $τ$ is underlined. Given a permutation $σ$, we say that $σ$ has a $τ$-match at position $i$ if $τ$ occurs in $σ$ in such a way that $σ_i$ plays the role of the underlined element in the occurrence. We let $pmp_τ(σ)$ denote the number of positions $i$ which $σ$ has a $τ$-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns $1\underline{2}3$ and $1\underline{3}2$ give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several collections of patterns like $\left \{1\underline{2}3 , 1\underline{3}2 \right \}$, and $\left \{ 1\underline234, 1\underline243, \underline2134, \underline2 1 4 3 \right \}$, as well as two positional marked patterns of any length $n$: $\left \{ 1\underline 2τ, \underline 21τ\right \}$.

math.CO↗
arXiv2018

Quadrant marked mesh patterns in 123-avoiding permutations

Given a permutation $σ= σ_1 \ldots σ_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $σ_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $σ$ if there are at least $a$ points to the right of $σ_i$ in $σ$ which are greater than $σ_i$, at least $b$ points to the left of $σ_i$ in $σ$ which are greater than $σ_i$, at least $c$ points to the left of $σ_i$ in $σ$ which are smaller than $σ_i$, and at least $d$ points to the right of $σ_i$ in $σ$ which are smaller than $σ_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions.

math.CO↗
arXiv2018

Exploring a Delta Schur Conjecture

In \cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $Δ_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in \cite{GHRY} that the Delta Conjecture is true when either $q=0$ or $t=0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $Δ_{s_ν} e_n$ by the same methods developed in \cite{GHRY}. Our first need here is a method for constructing a symmetric function that may be viewed as a "combinatorial side" for the symmetric function $Δ_{s_ν} e_n$ for $t=0$. Based on what was discovered in \cite{GHRY} we conjectured such a construction mechanism. We prove here that in the case that $ν=(m-k,1^k)$ with $1\le m< n$ the equality of the two sides can be established by the same methods used in \cite{GHRY}. While this work was in progress, we learned that Rhodes and Shimozono had previously constructed also such a "combinatorial side". Very recently, Jim Haglund was able to prove that their conjecture follows from the results in \cite{GHRY}. We show here that an appropriate modification of the Haglund arguments proves that the polynomial $Δ_{s_ν}e_n$ as well as the Rhoades-Shimozono "combinatorial side" have a plethystic evaluation with hook Schur function expansion.

math.CO↗
arXiv2017

A proof of the Delta Conjecture when $q=0$

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 operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials [A. M. Garsia and M. Haiman. A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244], [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371-407]. The Delta Conjecture involves two parameters $q,t$; in this article we give the first proof that the Delta Conjecture is true when $q=0$ or $t=0$.

math.CO↗
arXiv2017

On the Schur positivity of $Δ_{e_2} e_n[X]$

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_λ[X]$ in $Δ_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_λ[X]$ in $Δ_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.

math.CO↗
arXiv2017

Paired patterns in lattice paths

Let $\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \in \{E,N\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \in \mathcal{L}_n$ and a subset $A$ of $[n] = \{1, \ldots, n\}$, we let $ps_{L}(A)$ denote the word that results from $L$ by removing the $i^{th}$ occurrence of $E$ and the $i^{th}$ occurrence of $N$ in $L$ for all $i \in [n]-A$, reading from left to right. Then we say that a paired pattern $P \in \mathcal{L}_k$ occurs in $L$ if there is some $A \subseteq [n]$ of size $k$ such that $ps_L(A) = P$. In this paper, we study the generating functions of paired pattern matching in $\mathcal L_n$.

math.CO↗
arXiv2017

Descent c-Wilf Equivalence

Let $S_n$ denote the symmetric group. For any $σ\in S_n$, we let $\mathrm{des}(σ)$ denote the number of descents of $σ$, $\mathrm{inv}(σ)$ denote the number of inversions of $σ$, and $\mathrm{LRmin}(σ)$ denote the number of left-to-right minima of $σ$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $α$ and $β$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $α$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $β$. We give many examples of pairs of permutations $α$ and $β$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $α$ and $β$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(α) = \mathrm{des}(β)$ and $\mathrm{inv}(α) = \mathrm{inv}(β)$, then $α$ and $β$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.

math.CO↗
arXiv2017

Generating functions for permutations which avoid consecutive patterns with multiple descents

Let $S_n$ denote the group all permutations of $n$. For every permutation $σ$, we let $\mathrm{des}(σ)$ denote the number of descents in $σ$ and $\mathrm{LRMin}(σ)$ denote the number of left-to-right minima of $σ$. Given a sequence $τ= τ_1 \cdots τ_n$ of distinct positive integers, we define the reduction of $τ$, $\mathrm{red}(τ)$, to be the permutation of $S_n$ that results by replacing the $i$-th smallest element of $τ$ by $i$. If $Γ$ is a set of permutations, we say that a permutation $σ= σ_1 \ldots σ_n \in S_n$ has a $Γ$-match starting at position $i$ if there is a $i < j$ such that $\mathrm{red}(σ_i σ_{i+1} \ldots σ_j) \in Γ$. We let $Γ$-$\mathrm{mch}(σ)$ denote the number of $Γ$-matches in $σ$. We let $\mathcal{NM}_n(Γ)$ be the set of $σ\in S_n$ such that $Γ$-$\mathrm{mch}(σ) = 0$. In this paper, we modify Jones and Remmel's reciprocity method to study the generating function of the form \begin{equation} \mbox{NM}_Γ(t,x,y)=\sum_{n \geq 0} \frac{t^n}{n!} \mbox{NM}_{Γ,n}(x,y) \end{equation} where $\displaystyle \mbox{NM}_{Γ,n}(x,y) =\sum_{σ\in \mathcal{NM}_n(Γ)}x^{\mathrm{LRmin}(σ)}y^{1+\mathrm{des}(σ)}$ in the case where we no longer insist that all the permutations $τ\in Γ$ have at most one descent.

math.CO↗
arXiv2017

Q-analogues of the Fibo-Stirling numbers

Let $F_n$ denote the $n^{th}$ Fibonacci number relative to the initial conditions $F_0=0$ and $F_1=1$. Bach, Paudyal, and Remmel introduced Fibonacci analogues of the Stirling numbers called Fibo-Stirling numbers of the first and second kind. These numbers serve as the connection coefficients between the Fibo-falling factorial basis $\{(x)_{\downarrow_{F,n}}:n \geq 0\}$ and the Fibo-rising factorial basis $\{(x)_{\uparrow_{F,n}}:n \geq 0\}$ which are defined by $(x)_{\downarrow_{F,0}} = (x)_{\uparrow_{F,0}} = 1$ and for $k \geq 1$, $(x)_{\downarrow_{F,k}} = x(x-F_1) \cdots (x-F_{k-1})$ and $(x)_{\uparrow_{F,k}} = x(x+F_1) \cdots (x+F_{k-1})$. We gave a general rook theory model which allowed us to give combinatorial interpretations of the Fibo-Stirling numbers of the first and second kind. There are two natural $q$-analogues of the falling and rising Fibo-factorial basis. That is, let $[x]_q = \frac{q^x-1}{q-1}$. Then we let $[x]_{\downarrow_{q,F,0}} = \overline{[x]}_{\downarrow_{q,F,0}} = [x]_{\uparrow_{q,F,0}} = \overline{[x]}_{\uparrow_{q,F,0}}=1$ and, for $k > 0$, we let $[x]_{\downarrow_{q,F,k}} = [x]_q [x-F_1]_q \cdots [x-F_{k-1}]_q$, $\overline{[x]}_{\downarrow_{q,F,k}}= [x]_q ([x]_q-[F_1]_q) \cdots ([x]_q-[F_{k-1}]_q)$, $[x]_{\uparrow_{q,F,k}}= [x]_q [x+F_1]_q \cdots [x+F_{k-1}]_q$, and $\overline{[x]}_{\uparrow_{q,F,k}}= [x]_q ([x]_q+[F_1]_q) \cdots ([x]_q+[F_{k-1}]_q)$. In this paper, we show we can modify the rook theory model of Bach, Paudyal, and Remmel to give combinatorial interpretations for the two different types $q$-analogues of the Fibo-Stirling numbers which arise as the connection coefficients between the two different $q$-analogues of the Fibonacci falling and rising factorial bases. \end{abstract}

math.CO↗
arXiv2016

A Fibonacci analogue of Stirling numbers

Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n =F_{n-1}+F_{n-2}$ for $n \geq 3$. We let $n_F! = F_1 \cdots F_n$ and $\binom{n}{k}_F = \frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\downarrow_0} = (x)_{\uparrow_0} = 1$ and for $k \geq 1$, $(x)_{\downarrow_k} = x(x-1) \cdots (x-k+1)$ and $(x)_{\uparrow_k} = x(x+1) \cdots (x+k-1)$. Then the Stirling numbers of the first and second kind are the connections coefficients between the usual power basis $\{x^n:n \geq 0\}$ and the falling factorial basis $\{(x)_{\downarrow_n}:n \geq 0\}$ in the polynomial ring $\mathbb{Q}[x]$ and the Lah numbers are the connections coefficients between the rising factorial basis $\{(x)_{\uparrow_n}:n \geq 0\}$ and the falling factorial basis $\{(x)_{\downarrow_n}:n \geq 0\}$ in the polynomial ring $\mathbb{Q}[x]$. The goal of this paper is to find Fibonacci analogues for the Stirling numbers of the first and second kind and the Lah numbers. Our idea is to replace the falling factorial basis and the rising factorial basis by the Fibo-falling factorial basis $\{(x)_{\downarrow_{F,n}}:n \geq 0\}$ and the Fibo-rising factorial basis $\{(x)_{\uparrow_{F,n}}:n \geq 0\}$ where $(x)_{\downarrow_{F,0}} = (x)_{\uparrow_{F,0}} = 1$ and for $k \geq 1$, $(x)_{\downarrow_{F,k}} = x(x-F_1) \cdots (x-F_{k-1})$ and $(x)_{\uparrow_{F,k}} = x(x+F_1) \cdots (x+F_{k-1})$. Then we study the combinatorics of the connection coefficients betweenthe usual power basis, the Fibo-falling factorial basis, and the Fibo-rising factorial basis. In each case, we can give a rook theory model for the connections coefficients and show how this rook theory model can give combinatorial explanations for many of the properties of these coefficients.

math.CO↗
arXiv2016

Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

A permutation $τ$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $τ$ in a permutation $σ$ can share at most one element. Bóna \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $σ$, we let $\text{Des}(σ)$ denote the set of descents of $σ$. We study the class of permutations $σ\in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $σ$ such that $\text{Des}(σ) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$.

math.CO↗
arXiv2015

Generating functions for descents over permutations which avoid sets of consecutive patterns

We extend the reciprocity method of Jones and Remmel to study generating functions of the form $$\sum_{n \geq 0} \frac{t^n}{n!} \sum_{σ\in \mathcal{NM}_n(Γ)}x^{\mathrm{LRmin}(σ)}y^{1+\mathrm{des}(σ)}$$ where $Γ$ is a set of permutations which start with 1 and have at most one descent, $\mathcal{NM}_n(Γ)$ is the set of permutations $σ$ in the symmetric group $\mathfrak{S}_n$ which have no $Γ$-matches, $\mathrm{des}(σ)$ is the number of descents of $σ$ and $\mathrm{LRmin}(σ)$ is the number of left-to-right minima of $σ$. We show that this generating function is of the form $\left( \frac{1}{U_Γ(t,y)}\right)^x$ where $U_Γ(t,y) = \sum_{n\geq 0}U_{Γ,n}(y) \frac{t^n}{n!}$ and the coefficients $U_{Γ,n}(y)$ satisfy some simple recursions in the case where $Γ$ equals $\{1324,123\}$, $\{1324 \cdots p,12 \cdots (p-1)\}$ for $p \geq 5$, or $Γ$ is the set of permutations $σ= σ_1 \cdots σ_n$ of length $n=k_1+k_2$ where $k_1,k_2 \geq 2$, $σ_1 =1$, $σ_{k_1+1}=2$, and $\mathrm{des}(σ) =1$.

math.CO↗
arXiv2014

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-Löf tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness.

cs.LO↗
arXiv2014

An extension of MacMahon's Equidistribution Theorem to ordered set partitions

We prove a conjecture of Haglund which can be seen as an extension of the equidistribution of the inversion number and the major index over permutations to ordered set partitions. Haglund's conjecture implicitly defines two statistics on ordered set partitions and states that they are equidistributed. The implied inversion statistic is equivalent to a statistic on ordered set partitions studied by Steingrímsson, Ishikawa, Kasraoui, and Zeng, and is known to have a nice distribution in terms of $q$-Stirling numbers. The resulting major index exhibits a combinatorial relationship between $q$-Stirling numbers and the Euler-Mahonian distribution on the symmetric group, solving a problem posed by Steingrímsson.

math.CO↗
arXiv2014

Block patterns in Stirling permutations

We introduce and study a new notion of patterns in Stirling and $k$-Stirling permutations, which we call block patterns. We prove a general result which allows us to compute generating functions for the occurrences of various block patterns in terms of generating functions for the occurrences of patterns in permutations. This result yields a number of applications involving, among other things, Wilf equivalence of block patterns and a new interpretation of Bessel polynomials. We also show how to interpret our results for a certain class of labeled trees, which are in bijection with Stirling permutations.

math.CO↗
arXiv2012

Expressing Preferences using Preference Set Constraint Atoms

This paper introduces an extension of Answer Set Programming called Preference Set Constraint Programming which is a convenient and general formalism to reason with preferences. PSC programming extends Set Constraint Programming introduced by Marek and Remmel (Marek and Remmel 2004) by introducing two types of preference set constraint atoms, measure preference set constraint atoms and pre-ordered preference set constraint atoms, which are extensions of set constraint atoms. We show that the question of whether a PSC program has a preferred stable model is CoNP-complete. We give examples of the uses of the preference set constraint atoms and show that Answer Set Optimization (Brewka, Niemelä, and Truszczynski 2003) and General Preference (Son and Pontelli 2006) can be expressed using preference set constraint atoms.

cs.LO↗
arXiv2010

Ranking and unranking trees with a given number or a given set of leaves

In this paper, we provide algorithms to rank and unrank certain degree-restricted classes of Cayley trees (spanning trees of the n-vertex complete graph). Specifically, we consider classes of trees that have a given set of leaves or a fixed number k of leaves. For fixed k, the number of Cayley trees with n vertices and k leaves grows roughly as n! and hence the ranks have O(nlog_2(n)) bits. Our ranking and unranking algorithms require at most O(n^2) comparisons of numbers less than or equal to n plus O(n) operations of multiplication, division, addition, substraction and comparision on numbers of length O(nlog(n)).

math.CO↗