Advances in Applied Mathematics最新文献

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Discrete orthogonal ensemble on the exponential lattices

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2025-01-03 DOI: 10.1016/j.aam.2024.102836

Shi-Hao Li , Bo-Jian Shen , Guo-Fu Yu , Peter J. Forrester

Inspired by Aomoto's q-Selberg integral, a study is made of an orthogonal ensemble on an exponential lattice. By introducing a skew symmetric kernel, the configuration space of this ensemble is constructed to be symmetric and thus the corresponding skew inner product, skew orthogonal polynomials as well as correlation functions are explicitly formulated. These involve polynomials from the Askey scheme. Examples considered include the Al-Salam & Carlitz, q-Laguerre, little q-Jacobi and big q-Jacobi cases.

引用次数: 0

Cells of fixed height in Catalan words and restricted growth functions

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-30 DOI: 10.1016/j.aam.2024.102835

Aubrey Blecher, Arnold Knopfmacher

A word

w = w_{1} w_{2} \dots w_{n}

of length n over the set of positive integers is called a Catalan word whenever

w_{1} = 1

and

1 \leq w_{k} \leq w_{k - 1} + 1

for

k = 2, 3, \dots, n

. A restricted growth function is defined as a word

w = w_{1} w_{2} \dots w_{n}

of length n over the set of positive integers where

w_{1} = 1

and for

k \geq 2

we have

1 \leq w_{k} \leq \max {w_{1}, w_{2}, \dots, w_{k - 1}} + 1

. We also define cells and heights of cells and we represent such words as bargraphs (otherwise known as polyominoes) where the ith column contains

w_{i}

cells for

1 \leq i \leq n

and where all columns have their bottom cell on the x-axis. In the case of Catalan words, we prove a relationship between the number of cells at different heights and first terms of the expanded polynomial

{(1 + x)}^{2 n}

. In the case of restricted growth functions we find polynomials

F_{n} (x)

where the coefficient of

x^{j}

counts the number of cells of height j across all rgfs with n parts. In this case we also find bivariate generating functions for rgfs with k blocks, where the generating functions tracks the number of cells at a given height as well as the number of parts.

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引用次数: 0

Conditions for virtually Cohen–Macaulay simplicial complexes

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-19 DOI: 10.1016/j.aam.2024.102830

Adam Van Tuyl , Jay Yang

A simplicial complex Δ is a virtually Cohen–Macaulay simplicial complex if its associated Stanley-Reisner ring S has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length

codim (S)

. We provide a sufficient condition on Δ to be a virtually Cohen–Macaulay simplicial complex. We also introduce virtually shellable simplicial complexes, a generalization of shellable simplicial complexes. Virtually shellable complexes have the property that they are virtually Cohen–Macaulay, generalizing the well-known fact that shellable simplicial complexes are Cohen–Macaulay.

引用次数: 0

Dynamics of pop-tsack torsing

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-18 DOI: 10.1016/j.aam.2024.102826

Anqi Li

<div><div>For a finite irreducible Coxeter group <math><mo>(</mo><mi>W</mi><mo>,</mo><mi>S</mi><mo>)</mo></math> with a fixed Coxeter element c and set of reflections T, Defant and Williams define a pop-tsack torsing operation <math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>:</mo><mi>W</mi><mo>→</mo><mi>W</mi></math> given by <math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>w</mi><mo>⋅</mo><mi>π</mi><msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> where <math><mi>π</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>⋁</mo></mrow><mrow><mi>t</mi><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub><mi>w</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><mrow><mi>N</mi><mi>C</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></msubsup><mi>t</mi></math> is the join of all reflections lying below w in the absolute order in the non-crossing partition lattice <math><mi>N</mi><mi>C</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>c</mi><mo>)</mo></math>. This is a “dual” notion of the pop-stack sorting operator <math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math>; <math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math> was introduced by Defant as a way to generalize the pop-stack sorting operator on <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> to general Coxeter groups. Define the forward orbit of an element <math><mi>w</mi><mo>∈</mo><mi>W</mi></math> to be <math><msub><mrow><mi>O</mi></mrow><mrow><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow></mrow></msub><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>w</mi><mo>,</mo><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>(</mo><mi>w</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>w</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>}</mo></math>. Defant and Williams established the length of the longest possible forward orbits <math><msub><mrow><mi>max</mi></mrow><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow></msub><mo>⁡</mo><mo>|</mo><msub><mrow><mi>O</mi></mrow><mrow><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow></mrow></msub><mo>(</mo><mi>w</mi><mo>)</mo><mo>|</mo></math> for Coxeter groups of coincidental types and Type D in terms of the corresponding Coxeter number of the group. In their paper, they also proposed multiple conjectures about enumerating elements with near maximal orbit length. We resolve all the

引用次数: 0

Letter frequency vs factor frequency in pure morphic words

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-17 DOI: 10.1016/j.aam.2024.102834

Shuo Li

We prove that, for any pure morphic word w, if the frequencies of all letters in w exist, then the frequencies of all factors in w exist as well. This result answers a question of Saari in his doctoral thesis.

引用次数: 0

Initial values of ML-degree polynomials

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-13 DOI: 10.1016/j.aam.2024.102831

Maciej Gałązka

We prove a conjecture about the initial values of ML-degree polynomials stated by Michałek, Monin, and Wiśniewski.

引用次数: 0

Chordal matroids arising from generalized parallel connections II

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-13 DOI: 10.1016/j.aam.2024.102833

James Dylan Douthitt, James Oxley

In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of

G F (q)

-chordal matroids as those matroids that can be constructed from projective geometries over

G F (q)

by a sequence of generalized parallel connections across projective geometries over

G F (q)

. Our main result showed that when

q = 2

, such matroids have no induced minor in

{M (C_{4}), M (K_{4})}

. In this paper, we show that the class of

G F (2)

-chordal matroids coincides with the class of binary matroids that have none of

M (K_{4})

M^{⁎} (K_{3, 3})

, or

M (C_{n})

for

n \geq 4

as a flat. We also show that

G F (q)

-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices.

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引用次数: 0

Consecutive pattern containment and c-Wilf equivalence

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-13 DOI: 10.1016/j.aam.2024.102829

Reza Rastegar

We offer elementary proofs for several results in consecutive pattern containment that were previously demonstrated using ideas from cluster method and analytical combinatorics. Furthermore, we establish new general bounds on the growth rates of consecutive pattern avoidance in permutations.

引用次数: 0

Simultaneous orderings and combinatorial interpretations of generalized factorials and binomial coefficients

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-12-05 DOI: 10.1016/j.aam.2024.102810

Lan Nguyen

In this paper, we answer several questions raised by Manjul Bhargava concerning p-orderings, p-sequences, and combinatorial interpretations of generalized factorials and binomial coefficients associated with subsets S of

Z

. First, we prove some results of Bhargava concerning p-orderings, p-sequences and the integrality of generalized binomial coefficients directly from definitions, answering two questions of Bhargava. As a result, we also obtain some explicit descriptions of the p-sequences associated with p-orderings which do not exist in Bhargava's proofs. Second, we provide some combinatorial interpretations for the generalized factorials and the generalized binomial coefficients associated with subsets of

Z

which possess simultaneous p-orderings, answering two other questions raised by Bhargava.

引用次数: 0

A direct proof of well-definedness for the polymatroid Tutte polynomial 多面体图特多项式定义明确性的直接证明

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics

Pub Date : 2024-11-27 DOI: 10.1016/j.aam.2024.102809

Xiaxia Guan , Xian'an Jin

For a polymatroid P over

[n]

, Bernardi et al. (2022) [1] introduced the polymatroid Tutte polynomial

T_{P}

relying on the order

1 < 2 < \dots < n

[n]

, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that

T_{P}

is equivalent to another polynomial that only depends on P. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.

Bernardi 等人（2022 年）[1] 根据 [n] 的阶 1<2<⋯<n，对 [n] 上的多母题 P 提出了多母题图特多项式 TP，它将经典的图特多项式从母题推广到多母题。在本文中，与 Tutte 利用边的阶数活动对所有生成树求和所定义的 Tutte 多项式的定义良好性的原始证明类似，我们给出了多马特人 Tutte 多项式定义良好性的直接而基本的证明。

引用次数: 0

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