The classic Cayley identity states that \det(\partial) (\det X)^s =s(s+1)...(s+n-1) (\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix ofindeterminates and \partial=(\partial/\partial x_{ij}) is the correspondingmatrix of partial derivatives. In this paper we present straightforwardcombinatorial proofs of a variety of Cayley-type identities, both old and new.The most powerful of these proofs employ Grassmann algebra (= exterior algebra)and Grassmann-Berezin integration. Among the new identities proven here are apair of "diagonal-parametrized" Cayley identities, a pair of"Laplacian-parametrized" Cayley identities, and the "product-parametrized" and"border-parametrized" rectangular Cayley identities.
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机译：经典的Cayley身份指出\ det（\ partial）（\ det X）^ s = s（s + 1）...（s + n-1）（\ det X）^ {s-1}其中X = （x_ {ij}）是一个不确定的n×n矩阵，\ partial =（\ partial / \ partial x_ {ij}）是偏导数的对应矩阵。在本文中，我们提供了各种新旧Cayley型恒等式的简单组合证明，其中最有力的证明是使用Grassmann代数（=外代数）和Grassmann-Berezin积分。在此证明的新身份中，有一对“对角参数化” Cayley身份，一对“拉普拉斯参数化” Cayley身份，以及“乘积参数化”和“边界参数化”矩形Cayley身份。
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