Báo cáo toán học: "A Combinatorial Proof of Andrews’ Smallest Parts Partition Function" ppsx

... of parts of λ is called the length of λ, denoted by l(λ). The weight of λ is the sum of parts, denoted by |λ|. We let P(n) denote the set of partitions of n. Let spt(n) denote the number of smallest ... give a combinatorial proof of Andrews’ smallest parts partition function with the aid of rooted partitions introduced by Chen and the author. 1 Introductio...

Ngày tải lên: 07/08/2014, 15:22

... electronic journal of combinatorics 11 (2004), #R9 3 about a proof described to him by N.M. Ferrers. The Ferrers shape of a partition is an array of boxes, left justified, in which the number of boxes in ... to the basics of partition theory, see [1]. 2 q-analogue of the Sum of Cubes In this section we introduce the following polynomial generalization of (1) along with a...

Ngày tải lên: 07/08/2014, 08:20

... to the area of log-concave sequences. Abstract We provide a combinatorial proof for the fact that for any fixed n, the sequence {i(n, k )} 0≤k ≤ ( n 2 ) of the numbers of permutations of length ... A Combinatorial Proof of the Log-concavity of a famous sequence counting permutations Mikl´os B´ona ∗ Submitted: Nov 24, ... 71–89. [3] P. A. MacMahon, The indices of permutati...

Ngày tải lên: 07/08/2014, 08:22

... (resp. right-hand) side of (2). The aim of this paper is to find a combinatorial proof of (2). In Section 2 we construct the sets F bi n of labeled bicolored forests on [n]andD n of certain labeled ... generalizations of |T n+1 | =(n +1) n and |P n+1 | =(n +1)!C n . Remark. In spite of the simple expressions, we have not proved (8), (11) and (12) in a bijective way. Also a di...

Ngày tải lên: 07/08/2014, 08:22

... published combinatorial proofs of the q-Pfaff-Saalsch¨utz identity. The main object of this paper is to provide a bijective proof of (1.5) by generalizing Zeilberger’s combinatorial proof of the ... in the framework of words, and in Section 3 the combinatorial proof of (1.8) using partitions of integers. Finally, in Section 4, we give a short proof of (1.3) from s...

Ngày tải lên: 07/08/2014, 08:22

... of n K 2 ’s. Thus we get the following nontrivia l property of trees which was observed by Bruns and Hibi [2]. the electronic journal of combinatorics 17 (2010), #R9 6 A combinatorial proof of ... gave another proof of (3). The main purpose of this paper is to prove (3) combinatorially. In the meanwhile, we get as corollaries the results of Bruns and Hibi [2] : a formula...

Ngày tải lên: 08/08/2014, 11:20

Ngày tải lên: 05/08/2014, 09:46

... more elegant formula, as a corollary of more general reciprocity theorems. Here, fol- lowing a suggestion of Goldman, we provide a direct combinatorial proof of this new formula. MR primary subject ... x)=(−1) d R(B; − x − 1). The existing proofs derive this as a corollary of other reciprocity theorems, but Gold- man [3] has suggested that a direct combinatorial proof ought to...

Ngày tải lên: 07/08/2014, 06:20

... first stage of our proof of Proposition 1.1 will use Proposition 2.1 The cardinality v n of the set V n satisfies the recurrence relation (1) subject to the given initial conditions. Proof. The initial ... the order of simple groups [9] and to the solutions of the Diophantine equation: x 2 +(x+1) 2 =y 2 [8]. We will obtain a new bijective proof for this recurrence in terms of t...

Ngày tải lên: 07/08/2014, 06:20

... number of classes in the partition is finite: The partition relation X → (Y 0 , ,Y m−1 ) n holds if the electronic journal of combinatorics 7 (2000), #R24 8 Proof. It is a well-known theorem of B. ... the proof given there could be modified to yield this slightly stronger result.) The proof given below first appeared in [6, Section 5, pp. 32–34]. (Actually, this is not quite true,...

Ngày tải lên: 07/08/2014, 06:20