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Register a new userA Cayley-type identity for trees
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Authors
Ran J. Tessler
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We prove a weighted generalization of the formula for the number of plane vertex-labeled trees.
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In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~ref{111015c} provides concrete examples of infinite series of CIM-groups.
A compound determinant identity for minors of rectangular matrices is established. As an application, we derive Vandermonde type determinant formulae for classical group characters.
We show that every connected $k$-chromatic graph contains at least $k^{k-2}$ spanning trees.
Let $mathcal{O}_n$ be the set of ordered labeled trees on ${0,...,n}$. A maximal decreasing subtree of an ordered labeled tree is defined by the maximal ordered subtree from the root with all edges being decreasing. In this paper, we study a new refinement $mathcal{O}_{n,k}$ of $mathcal{O}_n$, which is the set of ordered labeled trees whose maximal decreasing subtree has $k+1$ vertices.
We consider pairs of a set-valued column-strict tableau and a reverse plane partition of the same shape. We introduce algortithms for them, which implies a bijective proof for the finite sum Cauchy identity for Grothendieck polynomials and dual Grothendieck polynomials.
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