Báo cáo toán học: "A Bijective Proof of Borchardt’s Identity" docx

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Ngày tải lên: 05/08/2014, 09:46

... second coordinate (i.e., the number of ways of “placing k non-taking rooks on B”). It has long been known [5] that the rook numbers of a board B determine the rook numbers of the complementary board ... subset of [d] ì [d](where[d]isdenedtobe{1, 2, ,d})andtherook numbers r B k of a board are the number of subsets of B of size k such that no two elements have t...

Ngày tải lên: 07/08/2014, 06:20

... {i}.(Ingeneral, if f is a function and A is a set, then by f A we mean the image of A under f.That is, f A = {f (a) | a ∈ A} .) The failure of a partition relation is indicated by replacing the ... be), if an order does not have countable character, then it must have uncountable character. 1 We remark that an order X has countable character if and only if the negative...

Ngày tải lên: 07/08/2014, 06:20

... the q-Lagrange inversion theorems of Garsia [7] and Garsia-Remmel [9]. Garsia, Gessel and Stanton, and Singer have shown that Rogers-Ramanujan type identities may be derived by means of q-Lagrange ... and Zeilberger gave an alternative, much shorter proof of these identities in [5]. Zeilberger gave a q-Foata proof of the q-Pfa-Saalschăutz identity [20], inspired by Foatas...

Ngày tải lên: 07/08/2014, 06:22

... extension of the Littlewood-Richardson rule [Z]. Taking the specialization λ = , we obtain the decomposition of s à/ into Schur functions; it is simpler than the traditional formulation of the Littlewood-Richardson rule ... the same tableaux. Via the specialization ν = , we obtain yet another formulation of the Littlewood- Richardson rule in this case involving th...

Ngày tải lên: 07/08/2014, 06:23

... number of boxes in the first row is equal to the size of the first part, the number of boxes in the second row is equal to the size of the second part, etc. For example, the Ferrers diagram for the ... introduction to the basics of partition theory, see [1]. 2 q-analogue of the Sum of Cubes In this section we introduce the following polynomial generalizat...

Ngày tải lên: 07/08/2014, 08:20

... present a bijective proof of Borchardt’s identity, one which relies only on rearranging terms in a sum by means of sign-reversing involutions. The proof reveals interesting properties of pairs of permutations. ... sign-reversing involution of C(p, q). Theorem 3.8 implies that equation (2.3) is true, hence we have a bijective proof of Borchardt’s identity. the electro...

Ngày tải lên: 07/08/2014, 08:20

... an octahedron, though, one of the old vertices (a local height max) the electronic journal of combinatorics 11 (2004), #R61 9 A positive proof of the Littlewood-Richardson rule using the octahedron ... give a bijective proof of this, using the octahedron rule from [RR, P, FZ, S]. This bijection was first found by CW in the honeycomb model, where the...

Ngày tải lên: 07/08/2014, 08:20

... Subject Classifications: 0 5A0 5, 0 5A1 5 To Richard Stanley, who introduced me 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, ... A Combinatorial Proof of the Log-concavity of a famous sequence counting permutations Mikl´os B´ona ∗ Submitted: Nov 24, 2004; Accepted: Jan...

Ngày tải lên: 07/08/2014, 08:22

... electronic journal of combinatorics 11(2) (2005), #N3 3 A combinatorial pro of of Postnikov’s identity and a generalized enumeration of labeled trees Seunghyun Seo ∗ Department of Mathematics Brandeis ... left-hand (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...

Ngày tải lên: 07/08/2014, 08:22

... generalization of the famous q-Pfa-Saalschăutz identity. An elementary proofofthisidentityisalsogiven. 1 Introduction Throughout this paper we regard q as an indeterminate, and we follow the notation ... electronic journal of combinatorics 12 (2005), #N2 4 A Combinatorial Proof of a Symmetric q-Pfa-Saalschăutz Identity Victor J. W. Guo Center for Combinatorics, LPMC Nankai...

Ngày tải lên: 07/08/2014, 08:22

... 1986. the electronic journal of combinatorics 12 (2005), #R18 8 Theorem 2.3 (Aztec diamond theorem) The number of domino tilings of the Aztec diamond of order n is 2 n(n+1)/2 . Remark: The proof of ... between domino tilings of an Aztec diamond and non- intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of...

Ngày tải lên: 07/08/2014, 08:22

... and y. Then G is a 4 -regular graph, and G has a 1-factor con- taining e and a 1-factor avoiding e. According to Theorem 3, G has a k-factor containing e and a k-factor avoiding e for every k ... containing e and a 1-factor avoiding e, then G has a k-factor containing e and a k-factor avoiding e for every k ∈ {1, 2, . . . , r − 1}. Proof. Let F and F e be two...

Ngày tải lên: 07/08/2014, 13:21

... 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

... (2010), #R64 6 A Bijective Proof of a Major Index Theorem of Garsia and Gessel ∗ Mordechai Novi ck Department of Mathematics Hebrew University, Jerusalem, Israel mordecha.novick@mail.huji.ac.il Submitted: ... lemma we prove the theorem by establishing a bijection between shuffles of ordered sequences and a certain set of partitions. A special case of Ga...

Ngày tải lên: 08/08/2014, 12:22