A combinatorial identity for rooted labeled forests
نویسندگان
چکیده
منابع مشابه
The multinomial distribution on rooted labeled forests
For a probability distribution (ps; s 2 S) on a nite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a p-forest if given F has m edges the vector of out-degrees of vertices of F has a multinomial distribution with parameters m and (ps; s 2 S), and given also these out-degrees the distribution of F is uniform on all forests with the given out-...
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An irregular pyramid consists of a stack of successively reduced graphs. Each smaller graph is deduced from the preceding one using contraction or removal kernels. A contraction (resp. removal) kernel defines a forest of the initial (resp. dual ) graph, each tree of this forest being reduced to a single vertex (resp. dual vertex) in the reduced graph. A combinatorial map encodes a planar graph ...
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Erdos, P. L., A new bijection on rooted forests, Discrete Mathematics 111 (1993) 179-188. This paper extends the method due to Szekely and ErdBs (1989) on the enumeration of trees. A bijection is introduced on certain classes of rooted forests (more exactly, on the class of semilabelled forests). This method yields new easy proofs for some well-known theorems which use only elementary calculati...
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If F,G are two n×m matrices, then det(1+xFG) = ∑ P x |P det(FP )det(GP ) where the sum is over all minors [19]. An application is a new proof of the Chebotarev-Shamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. We can generalize this and show that det(1 + kL) is the number of rooted edge-k-colored spanning forests....
متن کاملA Family of Invariants of Rooted Forests
Abstract. Let A be a commutative k-algebra over a field of characteristic 0 and Ξ a linear operator defined on A. We first define a family of A-valued invariants Ψ for finite rooted forests by a recurrent algorithm using the operator Ξ. Then, we show that the invariant Ψ distinguishes rooted forests if (and only if) it distinguishes rooted trees T , and if (and only if) it distinguishes the qua...
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ژورنال
عنوان ژورنال: Aequationes mathematicae
سال: 2019
ISSN: 0001-9054,1420-8903
DOI: 10.1007/s00010-019-00662-9