The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $sum_{ngeq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations.
{"title":"The plethystic inverse of the odd Lie representations","authors":"S. Sundaram","doi":"10.1090/proc/15938","DOIUrl":"https://doi.org/10.1090/proc/15938","url":null,"abstract":"The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $sum_{ngeq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76984549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-24DOI: 10.22108/TOC.2020.116257.1631
T. Goy, M. Shattuck
In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more general form with the appropriate initial conditions, where r>2 is arbitrary.
{"title":"Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Number Entries","authors":"T. Goy, M. Shattuck","doi":"10.22108/TOC.2020.116257.1631","DOIUrl":"https://doi.org/10.22108/TOC.2020.116257.1631","url":null,"abstract":"In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more general form with the appropriate initial conditions, where r>2 is arbitrary.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"347 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79677711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-19DOI: 10.7494/OPMATH.2021.41.2.245
Mohammad R. Piri, S. Alikhani
We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $chi_{dom}^{prime}(G)$. We obtain some properties of $chi_{dom}^{prime}(G)$ and compute it for specific graphs. Also we examine the effects on $chi_{dom}^{prime}(G)$ when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.
{"title":"Introduction to dominated edge chromatic number of a graph","authors":"Mohammad R. Piri, S. Alikhani","doi":"10.7494/OPMATH.2021.41.2.245","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.2.245","url":null,"abstract":"We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $chi_{dom}^{prime}(G)$. We obtain some properties of $chi_{dom}^{prime}(G)$ and compute it for specific graphs. Also we examine the effects on $chi_{dom}^{prime}(G)$ when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82427733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-11DOI: 10.4310/JOC.2021.v12.n2.a8
Brian Hopkins, Hua Wang Saint Peter's University, Georgia Southern University
Agarwal introduced $n$-color compositions in 2000 and most subsequent research has focused on restricting which parts are allowed. Here we focus instead on restricting allowed colors. After three general results, giving recurrence formulas for the cases of given allowed colors, given prohibited colors, and colors satisfying modular conditions, we consider several more specific conditions, establishing direct formulas and connections to other combinatorial objects. Proofs are combinatorial, mostly using the notion of spotted tilings introduced by the first named author in 2012.
{"title":"Restricted color $n$-color compositions","authors":"Brian Hopkins, Hua Wang Saint Peter's University, Georgia Southern University","doi":"10.4310/JOC.2021.v12.n2.a8","DOIUrl":"https://doi.org/10.4310/JOC.2021.v12.n2.a8","url":null,"abstract":"Agarwal introduced $n$-color compositions in 2000 and most subsequent research has focused on restricting which parts are allowed. Here we focus instead on restricting allowed colors. After three general results, giving recurrence formulas for the cases of given allowed colors, given prohibited colors, and colors satisfying modular conditions, we consider several more specific conditions, establishing direct formulas and connections to other combinatorial objects. Proofs are combinatorial, mostly using the notion of spotted tilings introduced by the first named author in 2012.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"2014 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87838362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a Cayley graph of the elementary abelian $2$-group $mathbb{Z}_2^{n}$ with respect to a set $S$ of size $d$. We prove that for any such $G, S$ and $d$, the maximum degree of any induced subgraph of $G$ on any set of more than half the vertices is at least $sqrt d$. This is deduced from the recent breakthrough result of Huang who proved the above for the $n$-hypercube $Q^n$, in which the set of generators $S$ is the set of all vectors of Hamming weight $1$. Motivated by his method we define and study unitary signings of adjacency matrices of graphs, and compare them to the orthogonal signings of Huang. As a byproduct, we answer a recent question of Belardo et. al. about the spectrum of signed $5$-regular graphs.
{"title":"Unitary signings and induced subgraphs of Cayley graphs of $mathbb{Z}_2^{n}$","authors":"N. Alon, Kai Zheng","doi":"10.19086/AIC.17912","DOIUrl":"https://doi.org/10.19086/AIC.17912","url":null,"abstract":"Let $G$ be a Cayley graph of the elementary abelian $2$-group $mathbb{Z}_2^{n}$ with respect to a set $S$ of size $d$. We prove that for any such $G, S$ and $d$, the maximum degree of any induced subgraph of $G$ on any set of more than half the vertices is at least $sqrt d$. This is deduced from the recent breakthrough result of Huang who proved the above for the $n$-hypercube $Q^n$, in which the set of generators $S$ is the set of all vectors of Hamming weight $1$. Motivated by his method we define and study unitary signings of adjacency matrices of graphs, and compare them to the orthogonal signings of Huang. As a byproduct, we answer a recent question of Belardo et. al. about the spectrum of signed $5$-regular graphs.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"107 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80799410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant type formula analogous to the classical definition of Schur polynomials.
{"title":"Determinantal formulas for dual Grothendieck polynomials","authors":"A. Amanov, Damir Yeliussizov","doi":"10.1090/proc/16008","DOIUrl":"https://doi.org/10.1090/proc/16008","url":null,"abstract":"We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant type formula analogous to the classical definition of Schur polynomials.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"74 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82757442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $ngeq 2$. Second we exhibit for each $ngeq 1$ a natural homomorphism $psi:A_n(G)to mathcal{H}_n(G)$, where $mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $mathcal{H}_n(G)$ is nontrivial and $psi$ is surjective. It follows that for each $ngeq 1$ there are graphs $G$ for which $A_n(G)$ is nontrivial.
{"title":"Higher discrete homotopy groups of graphs","authors":"Bob Lutz","doi":"10.5802/ALCO.151","DOIUrl":"https://doi.org/10.5802/ALCO.151","url":null,"abstract":"This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $ngeq 2$. Second we exhibit for each $ngeq 1$ a natural homomorphism $psi:A_n(G)to mathcal{H}_n(G)$, where $mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $mathcal{H}_n(G)$ is nontrivial and $psi$ is surjective. It follows that for each $ngeq 1$ there are graphs $G$ for which $A_n(G)$ is nontrivial.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83774114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The classical Dehn--Sommerville relations assert that the $h$-vector of an Eulerian simplicial complex is symmetric. We establish three generalizations of the Dehn--Sommerville relations: one for the $h$-vectors of pure simplicial complexes, another one for the flag $h$-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric $h$-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of "errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relations.
{"title":"NON‐EULERIAN DEHN–SOMMERVILLE RELATIONS","authors":"Connor Sawaske, Lei Xue","doi":"10.1112/MTK.12072","DOIUrl":"https://doi.org/10.1112/MTK.12072","url":null,"abstract":"The classical Dehn--Sommerville relations assert that the $h$-vector of an Eulerian simplicial complex is symmetric. We establish three generalizations of the Dehn--Sommerville relations: one for the $h$-vectors of pure simplicial complexes, another one for the flag $h$-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric $h$-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of \"errors coming from the links.\" For simplicial complexes, this further extends Klee's semi-Eulerian relations.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73739729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
{"title":"Zeta functions of periodic cubical lattices and cyclotomic-like polynomials.","authors":"Y. Hiraoka, Hiroyuki Ochiai, T. Shirai","doi":"10.2969/aspm/08410093","DOIUrl":"https://doi.org/10.2969/aspm/08410093","url":null,"abstract":"Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84147286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.
{"title":"Dual Grothendieck polynomials via last-passage percolation","authors":"Damir Yeliussizov","doi":"10.5802/crmath.67","DOIUrl":"https://doi.org/10.5802/crmath.67","url":null,"abstract":"The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84979740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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