Combinatorial Proofs of Identities Involving Symmetric Matrices
نویسنده
چکیده
Brualdi and Ma found a connection between involutions of length n with k descents and symmetric k×k matrices with non-negative integer entries summing to n and having no row or column of zeros. From their main theorem they derived two alternating sums by algebraic means and asked for combinatorial proofs. The purpose of this note is to give such demonstrations.
منابع مشابه
Brick tabloids and the connection matrices between bases of symmetric functions
Egecioglu, ii. and J.B. Remmel, Brick tabloids and the connection matrices between bases of symmetric functions, Discrete Applied Mathematics 34 (1991) 107-120. Let H, denote the space of symmetric functions, homogeneous of degree n. In this paper we introduce a new set of combinatorial objects called I-brick tabloids and its variants, which we use *to give combinatorial interpretations of the ...
متن کاملPfaffian and Hafnian Identities in Shuffle Algebras
Chen’s lemma on iterated integrals implies that certain identities involving multiple integrals, such as the de Bruijn and Wick formulas, amount to combinatorial identities for Pfaffians and hafnians in shuffle algebras. We provide direct algebraic proofs of such shuffle identities, and obtain various generalizations. We also discuss some Pfaffian identities due to Sundquist and Ishikawa-Wakaya...
متن کاملCombinatorial proofs for some forest hook length identities
Chen, Gao and Guo gave in a recent paper many interesting identities involving hook lengths of trees and forests using an extension of Han’s expansion technique. We give combinatorial proofs of some of these identities.
متن کاملCombinatorial Proofs of Capelli's and Turnbull's identities from Classical Invariant Theory
0. Introduction. Capelli’s [C] identity plays a prominent role in Weyl’s [W] approach to Classical Invariant Theory. Capelli’s identity was recently considered by Howe [H] and Howe and Umeda [H-U]. Howe [H] gave an insightful representation-theoretic proof of Capelli’s identity, and a similar approach was used in [H-U] to prove Turnbull’s [T] symmetric analog, as well as a new anti-symmetric an...
متن کاملThe q-Binomial Theorem and two Symmetric q-Identities
We notice two symmetric q-identities, which are special cases of the transformations of 2φ1 series in Gasper and Rahman’s book (Basic Hypergeometric Series, Cambridge University Press, 1990, p. 241). In this paper, we give combinatorial proofs of these two identities and the q-binomial theorem by using conjugation of 2-modular diagrams.
متن کامل