Báo cáo toán học: "A Refinement of Cayley’s Formula for Trees" doc

... first formula. For the second formula, we note that L =1− K −1 , and a formula for the coefficients of K −1 follows from (8.4) and (5.10). Applying Lemma 2.5 to Theorem 8.3 gives hook length formulas ... of a vertex v in a forest is the number of descendants of v (including v). We will consider various kinds of trees and forests in this paper, and for each of them we will h...

Ngày tải lên: 07/08/2014, 08:22

... values of Tables 1 and 2 into Theorems 1.1 and 1.2 yields the summa- tion formulas for sums of m-th power for m =1, 2, ,11. In particular, for 1 ≤ m ≤ 5 we recover the formulas (1.2)–(1.6) of Warnaar ... existence of a general formula for S m,n (q), and left it as an open problem. It is the purpose of this paper to provide such a general formula, which turns out to be a q...

Ngày tải lên: 07/08/2014, 08:22

... noncombinatorial proof of (2) by a modified Riordan array theorem derived from theorems in Sprugnoli et al. [5]. Also we refine the formula for C β,γ (n) with aids of a Gould classes of inverse relation ... i  the electronic journal of combinatorics 15 (2008), #R52 5 A refinement of the formula for k-ary trees and the Gould-Vandermonde’s convolution Ricky X. F. Chen Center...

Ngày tải lên: 07/08/2014, 15:23

... evaluation reliability for a number of classifiers, which were generated as a part of a greedy search for kernel parameters and found some linear correlation between the two. This proof of concept is a ... 2006. Re-evaluating the role of BLEU in machine translation research. In The Proceedings of the Thirteenth Conference of the European Chapter of the Association for Compu...

Ngày tải lên: 08/03/2014, 02:21

... completed the proof of equation (1.4). 5 Proof of Equation 1.5 For each k ≥ 1letS k be the set of composite objects defined by S k = {(T,N):T ∈C k and N ∈N |T| }, where as before N is the set of height ... (u) n = u k for all k ≥ 1, where | u n means extract the coefficient of u n . The coefficients of f (u) n may be expressed in terms of the coefficients of F (u) by means of t...

Ngày tải lên: 07/08/2014, 06:22

... 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. ... of permutations. We will first give a brief history of this identity, indicating methods of proof. The permanent of a square matrix is the sum of its diagonal products: per(a...

Ngày tải lên: 07/08/2014, 08:20

... combinatorial explanation of a formula of A. Post- nikov relating bicolored rooted trees to bicolored binary trees. We also present gen- eralized formulas for the number of labeled k-ary trees, ... (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...

Ngày tải lên: 07/08/2014, 08:22

... the reduced form of T . pre-reduced form of T reduced form of T u-c-tree T Figure 5.1 Definition 7. Let G denote the set of all c-graphs. Let T 2 = {T 2 ∈ G | T 2 is a pre-reduced form of some genus ... 7.3 Proof. Let min(T) denote the minimum number of rotations needed for a u-c-graph T to rotate into itself. Since min(T ) <(the number of vertices of T ) fails for all...

Ngày tải lên: 07/08/2014, 15:23

... pages doi:10.1155/2008/717614 Research Article A Refinement of Jensen’s Inequality for a Class of Increasing and Concave Functions Ye Xia Department of Computer and Information Science and Engineering, University of Florida, Gainesville, ... number of cases, which we will show later. For n>2, the proof is nearly identical to that for part a. Let us suppose 1.6 has been...

Ngày tải lên: 22/06/2014, 03:20

... hypernym of X if X is a (kind of) Y. • Hyponym(Hypo): X is a hyponym of Y if X is a (kind of) Y. • Holonym(Holo): Y is a holonym of Y if X is a part of Y. • Meronym(Mero): X is a meronym of Y if ... them could lead to better per- formance. Table 5 shows the retrieval performance of combining both types of similarity functions for short keyword queries. The results for shor...

Ngày tải lên: 17/03/2014, 02:20