A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)...A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)which avoid patternsσ_(1),σ_(2),...,σ_(r).The aim of this paper is to investigate when,for certain patternsσ_(1),σ_(2),...,σ_(r),S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.We prove that for any{σ_(1),σ_(2),...,σ_(r)}?S_3,if{σ_(1),σ_(2),...,σ_(r)}is sign-balanced except for{132,213,231,312},then S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.In addition,we give some results in the case of avoiding some patterns of length 4.展开更多
研究具有几乎单型的区传递自同构群的区传递Steine 5-设计的分类问题。利用二齐次置换群分类定理证明了:一个具有几乎单型区传递自同构群G的Steiner 5-设计D,则要么是一个5-(12,6,1)设计且自同构群G≅M12或者是一个5-(24,8,1)且自同构群G...研究具有几乎单型的区传递自同构群的区传递Steine 5-设计的分类问题。利用二齐次置换群分类定理证明了:一个具有几乎单型区传递自同构群G的Steiner 5-设计D,则要么是一个5-(12,6,1)设计且自同构群G≅M12或者是一个5-(24,8,1)且自同构群G≅M24,要么设计的自同构群G的基柱Socle(G)只能是典型单群。The classification problem of block-transitive Steiner 5-designs with an almost simple block-transitive automorphism group is studied. By using the classification theorem of 2-homogeneous permutation groups, it is proved that if the Steiner 5-design has an almost simple block-transitive automorphism group, then either it is a 5-(12,6,1)design and the automorphism G≅M12or it is a 5-(24,8,1)design and the automorphism G≅M24, or the socle of the automorphism G, Socle(G)can only be a classical simple group.展开更多
Let N_(2)(m,n)denote the number of partitions of n without repeated odd parts whose M_(2)-rank is m and M_(2)(m,n)denote the number of partitions π of n with distinct odd parts and such that the partition has crank m...Let N_(2)(m,n)denote the number of partitions of n without repeated odd parts whose M_(2)-rank is m and M_(2)(m,n)denote the number of partitions π of n with distinct odd parts and such that the partition has crank m.In this paper,by means of some classical q-series identities,we prove some inequalities of N_(2)(m,n)and M_(2)(m,n).For example,for m≥4 and n≠2m+2,N_(2)(m,n)≤N_(2)(m,n+1).展开更多
In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as...In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12061030)the Natural Science Foundation of Hainan Province(Grant No.122RC652)2023 Excellent Science and Technology Innovation Team of Jiangsu Province Universities(Real-Time Industrial Internet of Things).
文摘A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)which avoid patternsσ_(1),σ_(2),...,σ_(r).The aim of this paper is to investigate when,for certain patternsσ_(1),σ_(2),...,σ_(r),S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.We prove that for any{σ_(1),σ_(2),...,σ_(r)}?S_3,if{σ_(1),σ_(2),...,σ_(r)}is sign-balanced except for{132,213,231,312},then S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.In addition,we give some results in the case of avoiding some patterns of length 4.
文摘研究具有几乎单型的区传递自同构群的区传递Steine 5-设计的分类问题。利用二齐次置换群分类定理证明了:一个具有几乎单型区传递自同构群G的Steiner 5-设计D,则要么是一个5-(12,6,1)设计且自同构群G≅M12或者是一个5-(24,8,1)且自同构群G≅M24,要么设计的自同构群G的基柱Socle(G)只能是典型单群。The classification problem of block-transitive Steiner 5-designs with an almost simple block-transitive automorphism group is studied. By using the classification theorem of 2-homogeneous permutation groups, it is proved that if the Steiner 5-design has an almost simple block-transitive automorphism group, then either it is a 5-(12,6,1)design and the automorphism G≅M12or it is a 5-(24,8,1)design and the automorphism G≅M24, or the socle of the automorphism G, Socle(G)can only be a classical simple group.
基金Supported by NSFC(No.12001309)Natural Science Foundation Youth Fund of Qinghai Province(No.2022-ZJ-972Q)。
文摘Let N_(2)(m,n)denote the number of partitions of n without repeated odd parts whose M_(2)-rank is m and M_(2)(m,n)denote the number of partitions π of n with distinct odd parts and such that the partition has crank m.In this paper,by means of some classical q-series identities,we prove some inequalities of N_(2)(m,n)and M_(2)(m,n).For example,for m≥4 and n≠2m+2,N_(2)(m,n)≤N_(2)(m,n+1).
基金Supported by Zhoukou Normal University High-Level Talents Start-Up Funds Research Project(Grant No.ZKNUC2022007)the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX240725).
文摘In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.
