منابع مشابه
Concave Compositions
Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function disco...
متن کاملOn flushed partitions and concave compositions
Abstract. In this work, we give combinatorial proofs for generating functions of two problems, i.e. flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution wi...
متن کاملCombinatorial proofs of Andrews’ formulas on concave compositions
Concave compositions were recently introduced by Andrews[3] in the study of orthogonal polynomials, see also Andrews [4]. A concave composition of even length 2m, is a sum of the form ∑ ai + ∑ bi such that a1 > a2 > · · · > am = bm < bm−1 < · · · < b1, where am ≥ 0, and all ai and bi are integers. Let CE(n) denote the set of concave compositions of even length that sum to n, and ce(n) be the ca...
متن کاملCompositions versus Cyclic Compositions
We prove that the sum of greatest common divisors of parts in all compositions of n equals the sum of lengths of all cyclic compositions of n. The proof highlights structural similarities between the set of compositions of n and the set of cyclic compositions of n. 1. Definitions and Statement A composition of the positive integer n is an ordered partition of n; that is, a vector ( ) r p p ...,...
متن کاملQuasi-concave Programming Quasi-concave Programming
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2011
ISSN: 1077-8926
DOI: 10.37236/2002