Pub Date : 2025-12-15DOI: 10.1016/j.jcta.2025.106152
M. Golafshan , M. Rigo , M.A. Whiteland
Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer to the number of k-binomial equivalence classes represented by its factors of length n.
The Thue–Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period .
We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs.
{"title":"Computing the k-binomial complexity of generalized Thue–Morse words","authors":"M. Golafshan , M. Rigo , M.A. Whiteland","doi":"10.1016/j.jcta.2025.106152","DOIUrl":"10.1016/j.jcta.2025.106152","url":null,"abstract":"<div><div>Two finite words are <em>k</em>-binomially equivalent if each subword (i.e., subsequence) of length at most <em>k</em> occurs the same number of times in both words. The <em>k</em>-binomial complexity of an infinite word is a function that maps the integer <span><math><mi>n</mi><mo>⩾</mo><mn>0</mn></math></span> to the number of <em>k</em>-binomial equivalence classes represented by its factors of length <em>n</em>.</div><div>The Thue–Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the <em>k</em>-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the <em>k</em>-binomial complexities of the 2-letter TM word. For larger alphabets, work by Lü, Chen, Wen, and Wu determined the 2-binomial complexity for <em>m</em>-letter TM words, for arbitrary <em>m</em>, but the exact behavior for <span><math><mi>k</mi><mo>⩾</mo><mn>3</mn></math></span> remained unresolved. They conjectured that the <em>k</em>-binomial complexity function of the <em>m</em>-letter TM word is eventually periodic with period <span><math><msup><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>.</div><div>We resolve the conjecture positively by deriving explicit formulae for the <em>k</em>-binomial complexity functions for any generalized TM word. We do this by characterizing <em>k</em>-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106152"},"PeriodicalIF":1.2,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-11DOI: 10.1016/j.jcta.2025.106154
Tewodros Amdeberhan , George E. Andrews , Cristina Ballantine
We consider relationships between classical Lambert series, multiple Lambert series and classical q-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.
{"title":"Lambert series and double Lambert series","authors":"Tewodros Amdeberhan , George E. Andrews , Cristina Ballantine","doi":"10.1016/j.jcta.2025.106154","DOIUrl":"10.1016/j.jcta.2025.106154","url":null,"abstract":"<div><div>We consider relationships between classical Lambert series, multiple Lambert series and classical <em>q</em>-series of the Rogers-Ramanujan type. We conclude with a contemplation on the Andrews-Dixit-Schultz-Yee conjecture.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106154"},"PeriodicalIF":1.2,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-05DOI: 10.1016/j.jcta.2025.106153
Yongjiang Wu , Zhiyi Liu , Lihua Feng , Yongtao Li
This paper resolves two open problems in extremal set theory. For a family and , we denote . The sturdiness is defined as the minimum over all . A family is called an IU-family if it satisfies the intersection constraint: for all , as well as the union constraint: for all . The well-known IU-Theorem states that every IU-family has size at most . In this paper, we prove that if is an IU-family, then . This confirms a recent conjecture proposed by Frankl and Wang.
As the second result, we establish a tight upper bound on the sum of sizes of cross t-intersecting separated families. Our result not only extends a previous theorem of Frankl, Liu, Wang and Yang on separated families, but also provides explicit counterexamples to an open problem proposed by them, thereby settling their problem in the negative.
{"title":"Two results on set families: Sturdiness and intersection","authors":"Yongjiang Wu , Zhiyi Liu , Lihua Feng , Yongtao Li","doi":"10.1016/j.jcta.2025.106153","DOIUrl":"10.1016/j.jcta.2025.106153","url":null,"abstract":"<div><div>This paper resolves two open problems in extremal set theory. For a family <span><math><mi>F</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span> and <span><math><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, we denote <span><math><mi>F</mi><mo>(</mo><mi>i</mi><mo>,</mo><mover><mrow><mi>j</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo><mo>=</mo><mo>{</mo><mi>F</mi><mo>﹨</mo><mo>{</mo><mi>i</mi><mo>}</mo><mo>:</mo><mi>F</mi><mo>∈</mo><mi>F</mi><mo>,</mo><mi>F</mi><mo>∩</mo><mo>{</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>}</mo><mo>=</mo><mo>{</mo><mi>i</mi><mo>}</mo><mo>}</mo></math></span>. The sturdiness <span><math><mi>β</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is defined as the minimum <span><math><mo>|</mo><mi>F</mi><mo>(</mo><mi>i</mi><mo>,</mo><mover><mrow><mi>j</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo><mo>|</mo></math></span> over all <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>. A family <span><math><mi>F</mi></math></span> is called an IU-family if it satisfies the intersection constraint: <span><math><mi>F</mi><mo>∩</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mo>∅</mo></math></span> for all <span><math><mi>F</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi></math></span>, as well as the union constraint: <span><math><mi>F</mi><mo>∪</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span> for all <span><math><mi>F</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi></math></span>. The well-known IU-Theorem states that every IU-family <span><math><mi>F</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span> has size at most <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span>. In this paper, we prove that if <span><math><mi>F</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></msup></math></span> is an IU-family, then <span><math><mi>β</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msup></math></span>. This confirms a recent conjecture proposed by Frankl and Wang.</div><div>As the second result, we establish a tight upper bound on the sum of sizes of cross <em>t</em>-intersecting separated families. Our result not only extends a previous theorem of Frankl, Liu, Wang and Yang on separated families, but also provides explicit counterexamples to an open problem proposed by them, thereby settling their problem in the negative.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106153"},"PeriodicalIF":1.2,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145689362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-05DOI: 10.1016/j.jcta.2025.106143
Shikang Yu , Tao Feng , Hengrui Liu
A complete mapping of a finite group G is a permutation such that is also a permutation of G. The complete mapping ϕ is called odd (resp. even) when ϕ is an odd (resp. even) permutation. To address whether the permutation groups generated by all complete mappings of a finite group G are “large” and primitive, Bors and Wang (2023) initiated the investigation into which groups admit both even and odd complete mappings. It is conjectured that such groups exist if and only if their Sylow 2-subgroups are trivial or noncyclic, except for some groups of small order. This paper reduces the problem of determining whether an arbitrary finite group admits both even and odd complete mappings to examining (1) the existence of complete mappings that stabilize a Sylow 2-subgroup for every finite simple group of Lie type and every sporadic simple group in the set , and (2) the existence of complete mappings that stabilize a subgroup of order (for some odd prime and ) for every Ree group with .
{"title":"Complete mappings stabilizing a subgroup and their parities","authors":"Shikang Yu , Tao Feng , Hengrui Liu","doi":"10.1016/j.jcta.2025.106143","DOIUrl":"10.1016/j.jcta.2025.106143","url":null,"abstract":"<div><div>A complete mapping of a finite group <em>G</em> is a permutation <span><math><mi>ϕ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></math></span> such that <span><math><mi>x</mi><mo>↦</mo><mi>x</mi><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is also a permutation of <em>G</em>. The complete mapping <em>ϕ</em> is called odd (resp. even) when <em>ϕ</em> is an odd (resp. even) permutation. To address whether the permutation groups generated by all complete mappings of a finite group <em>G</em> are “large” and primitive, Bors and Wang (2023) initiated the investigation into which groups admit both even and odd complete mappings. It is conjectured that such groups exist if and only if their Sylow 2-subgroups are trivial or noncyclic, except for some groups of small order. This paper reduces the problem of determining whether an arbitrary finite group admits both even and odd complete mappings to examining (1) the existence of complete mappings that stabilize a Sylow 2-subgroup for every finite simple group of Lie type and every sporadic simple group in the set <span><math><mo>{</mo><mrow><mi>Ly</mi></mrow><mo>,</mo><msub><mrow><mi>Co</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><mi>B</mi><mo>,</mo><mspace></mspace><mrow><mi>Th</mi></mrow><mo>,</mo><mspace></mspace><msub><mrow><mi>Fi</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>Fi</mi></mrow><mrow><mn>23</mn></mrow></msub><mo>,</mo><mspace></mspace><msubsup><mrow><mi>Fi</mi></mrow><mrow><mn>24</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mspace></mspace><mrow><mi>Ru</mi></mrow><mo>,</mo><msup><mrow><mi>O</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>N</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>Co</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>Co</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mrow><mi>HN</mi></mrow><mo>,</mo><mi>M</mi><mo>}</mo></math></span>, and (2) the existence of complete mappings that stabilize a subgroup of order <span><math><mn>8</mn><msubsup><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msubsup></math></span> (for some odd prime <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⩾</mo><mn>1</mn></math></span>) for every Ree group <span><math><mmultiscripts><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> with <span><math><mi>n</mi><mo>⩾</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106143"},"PeriodicalIF":1.2,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-01DOI: 10.1016/j.jcta.2025.106134
P. Pylyavskyy , S. Shirokovskikh , M. Skopenkov
We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff's matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon–D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.
{"title":"Superport networks","authors":"P. Pylyavskyy , S. Shirokovskikh , M. Skopenkov","doi":"10.1016/j.jcta.2025.106134","DOIUrl":"10.1016/j.jcta.2025.106134","url":null,"abstract":"<div><div>We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff's matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon–D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106134"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145650839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-28DOI: 10.1016/j.jcta.2025.106133
Jan Petr , Pavel Turek
Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and Pálvölgyi, we initiate the study of temperate families: a family is said to be temperate if each contains at most elements of as a proper subset.
We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube . As a more general result, we obtain that the middle layers of the hypercube maximise the size of a family such that each contains at most elements of as a proper subset. Moreover, we classify all such families consisting of the maximum number of sets.
In the case of intersecting temperate families, we find the maximum size and classify all intersecting temperate families consisting of the maximum number of sets for odd n. We also conjecture the maximum size for even n.
受Frankl、Pach和Pálvölgyi最近引入的奇向日葵的启发，我们发起了对温带家族的研究：如果每个a∈F最多包含F的|00个a |个元素作为适当子集，则称一个家族F∈P（[n]）是温带的。我们证明了温带族的最大尺寸是在超立方体{0,1}n的中间两层。作为一个更一般的结果，我们得到了超立方体的中间t+1层使族F的大小最大化，使得每个a∈F最多包含F的∑j=1t（| a |j）个元素作为一个固有子集。此外，我们对所有由最大数量集合组成的族进行了分类。对于相交的温带族，我们找到了最大的大小，并对奇数n下由最大集合数组成的所有相交的温带族进行了分类。我们还推测了偶数n下的最大大小。
{"title":"Towards odd-sunflowers: Temperate families and lightnings","authors":"Jan Petr , Pavel Turek","doi":"10.1016/j.jcta.2025.106133","DOIUrl":"10.1016/j.jcta.2025.106133","url":null,"abstract":"<div><div>Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and Pálvölgyi, we initiate the study of temperate families: a family <span><math><mi>F</mi><mo>⊆</mo><mi>P</mi><mo>(</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>)</mo></math></span> is said to be <em>temperate</em> if each <span><math><mi>A</mi><mo>∈</mo><mi>F</mi></math></span> contains at most <span><math><mo>|</mo><mi>A</mi><mo>|</mo></math></span> elements of <span><math><mi>F</mi></math></span> as a proper subset.</div><div>We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>. As a more general result, we obtain that the middle <span><math><mi>t</mi><mo>+</mo><mn>1</mn></math></span> layers of the hypercube maximise the size of a family <span><math><mi>F</mi></math></span> such that each <span><math><mi>A</mi><mo>∈</mo><mi>F</mi></math></span> contains at most <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mi>j</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> elements of <span><math><mi>F</mi></math></span> as a proper subset. Moreover, we classify all such families consisting of the maximum number of sets.</div><div>In the case of intersecting temperate families, we find the maximum size and classify all intersecting temperate families consisting of the maximum number of sets for odd <em>n</em>. We also conjecture the maximum size for even <em>n</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106133"},"PeriodicalIF":1.2,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145611784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-28DOI: 10.1016/j.jcta.2025.106132
Shi-Mei Ma , Jun-Ying Liu , Jean Yeh , Yeong-Nan Yeh
It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by many researchers. This paper is divided into three parts. The first part gives a convolution formula for the second-order Eulerian polynomials, which simplifies a result of Gessel. As an application, a determinantal expression for the second-order Eulerian polynomial is obtained. We then investigate a convolution formula of the trivariate second-order Eulerian polynomials. Among other things, by introducing three new statistics: proper ascent-plateau, improper ascent-plateau and trace, we discover that a six-variable enumerative polynomial over restricted Stirling permutations equals a six-variable Eulerian-type polynomial over signed permutations. By special parametrizations, we make use of Stirling permutations to give a unified interpretation of the -Eulerian polynomials and derangement polynomials of types A and B. The third part presents a box sorting algorithm which leads to a bijection between the terms in the expansion of and ordered weak set partitions, where c is a smooth function in the indeterminate x and D is the derivative with respect to x. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of in terms of standard Young tableaux. Combining this with context-free grammars, we provide three new interpretations of the second-order Eulerian polynomials.
{"title":"Eulerian-type polynomials over Stirling permutations and box sorting algorithm","authors":"Shi-Mei Ma , Jun-Ying Liu , Jean Yeh , Yeong-Nan Yeh","doi":"10.1016/j.jcta.2025.106132","DOIUrl":"10.1016/j.jcta.2025.106132","url":null,"abstract":"<div><div>It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by many researchers. This paper is divided into three parts. The first part gives a convolution formula for the second-order Eulerian polynomials, which simplifies a result of Gessel. As an application, a determinantal expression for the second-order Eulerian polynomial is obtained. We then investigate a convolution formula of the trivariate second-order Eulerian polynomials. Among other things, by introducing three new statistics: proper ascent-plateau, improper ascent-plateau and trace, we discover that a six-variable enumerative polynomial over restricted Stirling permutations equals a six-variable Eulerian-type polynomial over signed permutations. By special parametrizations, we make use of Stirling permutations to give a unified interpretation of the <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-Eulerian polynomials and derangement polynomials of types <em>A</em> and <em>B</em>. The third part presents a box sorting algorithm which leads to a bijection between the terms in the expansion of <span><math><msup><mrow><mo>(</mo><mi>c</mi><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>c</mi></math></span> and ordered weak set partitions, where <em>c</em> is a smooth function in the indeterminate <em>x</em> and <em>D</em> is the derivative with respect to <em>x</em>. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of <span><math><msup><mrow><mo>(</mo><mi>c</mi><mi>D</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>c</mi></math></span> in terms of standard Young tableaux. Combining this with context-free grammars, we provide three new interpretations of the second-order Eulerian polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106132"},"PeriodicalIF":1.2,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145611846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-21DOI: 10.1016/j.jcta.2025.106130
Jean-Christophe Novelli, Jean-Yves Thibon
As shown in our paper (Novelli and Thibon, 2021 [20]), the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to WQSym, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behavior with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-t version of these noncommutative analogues. For rectangular partitions, their commutative images at appear to coincide with the multi-t Hall-Littlewood functions introduced in (Lascoux et al., 1995 [14]). This leads us to conjecture that for rectangular partitions, multi-t Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.
如我们的论文（Novelli and Thibon, 2021[20]）所示，Shareshian-Wachs的色拟对称函数可以提升到WQSym，即非交换变量中拟对称函数的代数。我们在这里研究了它在经典字母变换中的行为，并提出了一个与Haglund-Wilson公式的非交换版本兼容的Macdonald多项式的非交换模拟。我们还介绍了这些非交换类似物的多t版本。对于矩形分区，它们在q=0处的交换像似乎与（Lascoux et al., 1995[14]）中引入的多t Hall-Littlewood函数一致。这使我们推测，对于矩形分区，多重t Macdonald多项式可以作为Hecke代数的某些Yang-Baxter元素的等变迹得到。我们还推测所有（普通）麦克唐纳多项式都可以用这种方法得到。最后，我们对拟对称色函数的各个方面与Hecke代数中的计算作了一些评述。特别地，我们证明了所有模关系都是由Kazhdan-Lusztig基的乘积公式给出的。
{"title":"Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation","authors":"Jean-Christophe Novelli, Jean-Yves Thibon","doi":"10.1016/j.jcta.2025.106130","DOIUrl":"10.1016/j.jcta.2025.106130","url":null,"abstract":"<div><div>As shown in our paper (Novelli and Thibon, 2021 <span><span>[20]</span></span>), the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to <strong>WQSym</strong>, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behavior with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-<em>t</em> version of these noncommutative analogues. For rectangular partitions, their commutative images at <span><math><mi>q</mi><mo>=</mo><mn>0</mn></math></span> appear to coincide with the multi-<em>t</em> Hall-Littlewood functions introduced in (Lascoux et al., 1995 <span><span>[14]</span></span>). This leads us to conjecture that for rectangular partitions, multi-<em>t</em> Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106130"},"PeriodicalIF":1.2,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145567380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-21DOI: 10.1016/j.jcta.2025.106131
Yunjing Shan, Junling Zhou
There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptography and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let V be an n-dimensional vector space over the finite field and let denote the family of all k-dimensional subspaces of V. A family is called cover-free if there are no three distinct subspaces such that . A family is called a q-Steiner system if for every , there is exactly one such that . In this paper we investigate cover-free families in the vector space V. Firstly, we determine the maximum size of a cover-free family in . Secondly, we characterize the structures of all maximum cover-free families which are closely related to q-Steiner systems.
{"title":"On cover-free families of finite vector spaces","authors":"Yunjing Shan, Junling Zhou","doi":"10.1016/j.jcta.2025.106131","DOIUrl":"10.1016/j.jcta.2025.106131","url":null,"abstract":"<div><div>There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptography and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let <em>V</em> be an <em>n</em>-dimensional vector space over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and let <span><math><msub><mrow><mo>[</mo><mfrac><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> denote the family of all <em>k</em>-dimensional subspaces of <em>V</em>. A family <span><math><mi>F</mi><mo>⊆</mo><msub><mrow><mo>[</mo><mfrac><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> is called cover-free if there are no three distinct subspaces <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>F</mi></math></span> such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≤</mo><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. A family <span><math><mi>H</mi><mo>⊆</mo><msub><mrow><mo>[</mo><mfrac><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> is called a <em>q</em>-Steiner system <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> if for every <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mo>[</mo><mfrac><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span>, there is exactly one <span><math><mi>H</mi><mo>∈</mo><mi>H</mi></math></span> such that <span><math><mi>T</mi><mo>≤</mo><mi>H</mi></math></span>. In this paper we investigate cover-free families in the vector space <em>V</em>. Firstly, we determine the maximum size of a cover-free family in <span><math><msub><mrow><mo>[</mo><mfrac><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Secondly, we characterize the structures of all maximum cover-free families which are closely related to <em>q</em>-Steiner systems.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106131"},"PeriodicalIF":1.2,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145567756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jcta.2025.106129
Ingo Czerwinski, Alexander Pott
A (binary) Sidon set M is a subset of such that the sum of four distinct elements of M is never 0. The goal is to find Sidon sets of large size. In this note we show that the graphs of almost perfect nonlinear (APN) functions with high linearity can be used to construct large Sidon sets. Thanks to recently constructed APN functions with high linearity, we can construct Sidon sets of size 192 in , where the largest sets so far had size 152. Using the inverse and the Dobbertin function also gives larger Sidon sets as previously known. Each of the new large Sidon sets M in yields a binary linear code with t check bits, minimum distance 5, and a length not known so far. Moreover, we improve the upper bound for the linearity of arbitrary APN functions.
{"title":"On large Sidon sets","authors":"Ingo Czerwinski, Alexander Pott","doi":"10.1016/j.jcta.2025.106129","DOIUrl":"10.1016/j.jcta.2025.106129","url":null,"abstract":"<div><div>A (binary) Sidon set <em>M</em> is a subset of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> such that the sum of four distinct elements of <em>M</em> is never 0. The goal is to find Sidon sets of large size. In this note we show that the graphs of almost perfect nonlinear (APN) functions with high linearity can be used to construct large Sidon sets. Thanks to recently constructed APN functions <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>8</mn></mrow></msubsup><mo>→</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>8</mn></mrow></msubsup></math></span> with high linearity, we can construct Sidon sets of size 192 in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>15</mn></mrow></msubsup></math></span>, where the largest sets so far had size 152. Using the inverse and the Dobbertin function also gives larger Sidon sets as previously known. Each of the new large Sidon sets <em>M</em> in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msubsup></math></span> yields a binary linear code with <em>t</em> check bits, minimum distance 5, and a length not known so far. Moreover, we improve the upper bound for the linearity of arbitrary APN functions.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"220 ","pages":"Article 106129"},"PeriodicalIF":1.2,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145559892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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