On the number of factorizations of a full cycle

We give a new expression for the number of factorizations of a full cycle into an ordered product of permutations of specified cycle types. This is done through purely algebraic means, extending recent work of Biane. We deduce from our result a remarkable formula of Poulalhon and Schaeffer that was previously derived through an intricate combinatorial argument.     

Minimal factorizations of permutations into star transpositions

We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form . This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner.     

A bijective proof of the hook-length formula for skew shapes

Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this paper, we present a simple bijection that proves an equivalent recursive version of Naruse’s result, in the same way that the celebrated hook-walk proof due to Greene, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary shapes.In particular, we also give a new bijective proof of the classical hook-length formula, quite different from the known proofs.     

A combinatorial proof of a Weyl type formula for hook Schur polynomials

In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which was obtained previously by other people using a Kostant type cohomology formula for . In general, we can obtain in a combinatorial way a Weyl type character formula for various irreducible highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair. This research was supported by 2007 research fund of University of Seoul.     

Hook,line and sinker: A bijective proof of the skew shifted hook-length formula

A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall.Recently, the formula for skew straight shapes was proved by the author via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is a bijection that proves the whole family of hook-length formulas, and is also the simplest known bijective proof for shifted (non-skew) shapes. The complexity of the algorithm is studied, and a weighted generalization of Naruse’s formula is also presented.     

A bijective proof of the generating function for the number of reverse plane partitions via lattice paths

Recently, Hillman and Grassl gave a bijective proof for the generating function for the number of reverse plane partitions of a fixed shape λ. We give another bijective proof for this generating function via completelv different methods. Our bijection depends on a lattice path coding of reverse plane partitions and a new method for constructing bisections out of certain pairs of involutions due to Garsia and Milne.     

A bijective proof of the generating function for the number of reverse plane partitions via lattice paths

Recently, Hillman and Grassl gave a bijective proof for the generating function for the number of reverse plane partitions of a fixed shape λ. We give another bijective proof for this generating function via completelv different methods. Our bijection depends on a lattice path coding of reverse plane partitions and a new method for constructing bisections out of certain pairs of involutions due to Garsia and Milne.     

A short proof for the sum formula and its generalization

For positive integers with a _r  = 2, the multiple zeta value or r-fold Euler sum is defined as [2]

Pieri’s formula for generalized Schur polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.     

A combinatorial proof of a formula for the Lucas-Narayana polynomials

In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for $1\le k\le n$ and for $n\ge 1$. In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers $N_{n,k,F}$, providing a combinatorial proof of the conjecture that these are positive integers for $n\ge 1$.     

A（x）－C（x）的渐近公式

设Ａ、Ｃ是一些自然数的集合。对于Ａ中任一自然数ｍ，每一ｍ阶群都是Ａｂｅｌ群；对于Ｃ中任一自然数ｎ，每一ｎ阶群都是循环群。本文的目的是证明下面的渐近公式：此处γ是Ｅｕｌｅｒ常数，ｌｏｇｒｘ＝ｌｏｇ（ｌｏｇ＿（ｒ－１）ｘ），ｌｏｇ＿１ｘ＝ｌｏｇｘ。     

A design and a geometry for the group Fi 22

The Fischer group Fi ₂₂ acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it.      

Minor summation formula and a proof of Stanley’s open problem

In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.      

A formula for the Whitehead group of a three-dimensional crystallographic group

We give an explicit formula for the Whitehead group of a three-dimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ.     

A simple formula for the number of spanning trees of line graphs

Suppose is a loopless graph and is the graph obtained from G by subdividing each of its edges k () times. Let be the set of all spanning trees of G, be the line graph of the graph and be the number of spanning trees of . By using techniques from electrical networks, we first obtain the following simple formula: Then we find it is in fact equivalent to a complicated formula obtained recently using combinatorial techniques in [F. M. Dong and W. G. Yan, Expression for the number of spanning trees of line graphs of arbitrary connected graphs, J. Graph Theory. 85 (2017) 74–93].     

The Bass-Heller-Swan formula for the equivariant topological Whitehead group

In the first part of this paper, a geometric definition of theK-theory equivariant nilpotent groups is given. For a finite groupG, the Nil-groups are defined as functors from the category ofG-spaces andG-homotopy classes ofG-maps to Abelian groups. In the nonequivariant case, these groups are isomorphic to the classical algebraic Nil-groups.In the second part, the Bass-Heller-Swan formula is proved for the equivariant topological Whitehead group. The main result of this work is that ifX is a compactG-ANR andG acts trivially onS ¹, then