Báo cáo toán học: "Lattice Paths between Diagonal Boundaries" pot

... and that there is a natural bijection between these permutations and lattice paths from (0, 0) to (j, j) using steps (0, 1) and (1, 0) which never go above the main diagonal. Also, let C(x)=  n≥0 c n x n = 1− √ 1−4x 2x . ... diagonal (say it stays below). Suppose it touches the diagonal first at the point (i, i). If there is an entry of rank more than 1 inserted between the ith an...

Ngày tải lên: 07/08/2014, 06:22

... lattice paths, d[m, n]=d[m−1,n]+d[m, n − 1] + γd[m − 1,n−1]. The paths take horizontal, vertical, and diagonal steps (a king). If we write γ = µ − 1, then this recurrence enumerates lattice paths ... Brownian motion between parallel reflecting walls, a continuous version equivalent to the asymptotic distribution of the Kolmogorov-Smirnov test. The exact number of ordinary paths bet...

Ngày tải lên: 07/08/2014, 06:22

... weighted lattice paths can be interpreted as probability distributions arising in the context of P´olya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well ... n/(m + n). The considered lattice paths are absorbed at lines y = x/t − s/t with t ∈ N and s ∈ N 0 . We provide explicit formulæ for the sum of the weights of paths, starting at (m, n), which...

Ngày tải lên: 07/08/2014, 21:21

Ngày tải lên: 05/08/2014, 15:20

... over rings, co -diagonal matrices 1 Introduction In this work we examine matrices over a ring R, such that the diagonal elements of the matrix are all 0’s, but the elements off the diagonal are not ... diagonal are not zero (we shall call these matrices co -diagonal over R). We define the rank of a matrix over a ring, and show that low rank co- diagonal matrices over Z 6 naturally cor...

Ngày tải lên: 07/08/2014, 06:20

... generating function  n≥0 c n,k n! x n = d  ν=1 I k ν (2x). (2) 2.2 Connection with permutations without long ascending subse- quences The connection between lattice walks and the class of permutations that we are studying here is obtained by comparing ... the the electronic journal of combinatorics 5 (1998), #R2 10 Evidently there is a 1-1 correspondence between restricted growth functi...

Ngày tải lên: 07/08/2014, 06:22

... permutation of {1, ,d} and if instead of the matrix A we have the matrix A  whose i-th row has 1 on the diagonal, −δ σi at column σi and 0 elsewhere, we get again a lattice tiling with the lattice (A  )  Z d .Stein[St] as

Ngày tải lên: 07/08/2014, 06:22

... weight-preserving bijection between Schr¨oder paths and Schr¨oder permutations, Ann. Combin., 6 (2002), 235–248. [3] J. Bandlow and K. Killpatrick, An area-to-inv bijection Dyck paths and 312-avoiding ... corre- spondence with Schr¨oder paths. Dyck paths are also closely related to permutations with forbidden patterns, see Stanley [23], Krattenthaler [16] and West [26]. Motzkin path...

Ngày tải lên: 07/08/2014, 07:21

... 13 we have established a bijection between α-orientations and α-potentials. The following lemma completes the proof of Theorem 1. Lemma 14. The set of all α-potentials ℘ : E→IN with the dominance ... special case of this bijection is Temperley’s bijection between spanning trees and matchings of square grids. The general correspondence between trees and matchings has been exploited in [6...

Ngày tải lên: 07/08/2014, 08:20

... right of λ. We calculate generating functions for standard paths of fixed width and for standard paths of height ≤ 2. 1 Definition of standard paths By a composition P we mean a sequence of positive ... tableau is 421 3 . Clearly, two different standard paths give rise to different tableau. Furthermore, the tableau that occurs as tableau of standard paths must be increasing in every col...

Ngày tải lên: 07/08/2014, 08:22