arXiv subjects

Arthur L. B. Yang

Publications and source records attributed to Arthur L. B. Yang.

44 records · Page 3Linked to original sources

arXiv2008

The srank Conjecture on Schur's $Q$-Functions

We show that the shifted rank, or srank, of any partition $λ$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_λ$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition $λ/μ$ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank.

math.CO↗

arXiv2007

A Blass-Sagan bijection on Eulerian equivalence classes

Following the treatment of Blass and Sagan, we present an algorithmic bijection between the Eulerian equivalence classes of totally cyclic orientations and the spanning trees without internal activity edges for a given graph.

math.CO↗

arXiv2007

A major index for matchings and set partitions

We introduce a statistic $\pmaj$ on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inversion number and major index.

math.CO↗

arXiv2005

Stanley's Zrank Problem on Skew Partitions

We present an affirmative answer to Stanley's zrank problem, namely, the zrank and rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs depend on the number of zero entries. Similar to notion of the jrank of a skew partition, we give a characterization of the rank in terms of the Giambelli type matrices of the corresponding skew Schur functions. We also show that the sign of the determinant of a factorial Cauchy matrix is uniquely determined by the number of its zero entries, which implies the nonsingularity of the inverse binomial coefficient matrix.

math.CO↗

arXiv2005

Stable Equivalences of Giambelli Type Matrices of Schur Functions

By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This gives an affirmative answer to an open problem posed by Kuperberg.

math.CO↗

arXiv2005

Division and the Giambelli Identity

Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function, which generalize the Giambelli identity.

math.CO↗

arXiv2005

The Zrank Conjecture and Restricted Cauchy Matrices

The rank of a skew partition $λ/μ$, denoted $rank(λ/μ)$, is the smallest number $r$ such that $λ/μ$ is a disjoint union of $r$ border strips. Let $s_{λ/μ}(1^t)$ denote the skew Schur function $s_{λ/μ}$ evaluated at $x_1=...=x_t=1, x_i=0$ for $i>t$. The zrank of $λ/μ$, denoted $zrank(λ/μ)$, is the exponent of the largest power of $t$ dividing $s_{λ/μ}(1^t)$. Stanley conjectured that $rank(λ/μ)=zrank(λ/μ)$. We show the equivalence between the validity of the zrank conjecture and the nonsingularity of restricted Cauchy matrices. In support of Stanley's conjecture we give affirmative answers for some special cases.

math.CO↗

arXiv2004

Transformations of Border Strips and Schur Function Determinants

We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions including the Jacobi-Trudi determinant, its dual, the Giambelli determinant and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside decompositions of a given skew shape. Moreover, one can define the basic transformations which we call the twist transformation among cutting strips, and derive a transformation theorem for the determinantal formula of Hamel and Goulden. The special case of the transformation theorem for the Giambelli identity and the rim ribbon identity was obtained by Lascoux and Pragacz. Our transformation theorem also applies to the supersymmetric skew Schur function.

math.CO↗