Báo cáo toán học: "Counting set systems by weight" doc

... k, l. Representing set systems by incidence matrices, rows standing for edges and columns for the electronic journal of combinat orics 12 (2005), #R11 6 Counting set systems by weight Martin Klazar Institute ... problem for general set systems. We investigate these numbers in [8] and denote them h  n .Byh  n we denote the num- bers of vertex-labeled set systems with n i...

Ngày tải lên: 07/08/2014, 08:22

... step sets which are labeled arbitrarily by S i for i = 1 6. Table 1 contains a list of these step sets and an index to the path sets considered. Given a specific step set S i ,letA i (n)denotethesetofall ... path in L 2 (n) by independently coloring its peaks with colors from a set of z colors where we require blue (b)andred(r)tobe present whenever z ≥ 2. Let L c 2 (n; z) denote the s...

Ngày tải lên: 07/08/2014, 06:20

... defined in (4). Proof. Let M(n)bethesetofsimplep-free hypergraphs with the vertex set [n]and let n>1. We replace each H∈M(n) by a hypergraph H  with the vertex set [m], m = n/2 as follows. For ... are counted by the Schr¨oder numbers ([23], exercise 6.39.b) and it is easy to give a proof by GF; we omit the details. The extremal values of e(H)andi(H) follow from the formulas by...

Ngày tải lên: 07/08/2014, 06:20

... corresponding canonical sequences. The set of all partitions of [n] with exactly k blocks will be denoted by P (n, k) and has cardinality given by the Stirling number o f the second kind, denoted by S n,k . Fo r positive ... #R39. [7] M. K la zar, On abab-free and abba-free set part itio ns, Europ. J. Combin. 17 (1996) 53–68. [8] T. Mansour and A. Munagi, Enumeration of partitions...

Ngày tải lên: 08/08/2014, 11:20

... (2011), #P14 10 of blocks o f a set partition π is denoted by |π| and the set of all set partitions of X by Π(X). The number of set partitio ns of a n n-element set with exactly k blocks is called the ... empty set with no blocks, namely ∅. Let σ, π ∈ Π(X) and set σ ≤ π if every block of σ is a subset of a block in π, then (Π(X), ≤) becomes a poset. The maximal element ˆ 1 o...

Ngày tải lên: 08/08/2014, 12:23

... a m−2 a m−1 a m a m+1 a m+2 . . . a 2m−1 w = b 0 b 1 . . . b m−2 b m−1 b m . . . b 2m−3 We denote by W h,k (2m) the set of partial words w  of length 2m with h holes over a k-letter alphabet A such that, ... border of length l (respectively, a minimal border of length l), and by S h,k (n, l) and S  h,k (n, l) respectively the sets of such words. the electronic journal of combinatorics...

Ngày tải lên: 08/08/2014, 14:23

... is p  m=1 N m (−1) m+1 , where N m is the sum, over all m-subsets of the first p positions, of the number of words σ with k’s in the positions given by the subset. The quantity N m is given by N m =  p m  f r (n − m, k), since ... n ≥ p + q, we recursively count σ ∈ W [k] n (1 ′ 2 p,q ; r) by first counting σ such that at least one of the first p letters is k. By the principle of incl...

Ngày tải lên: 08/08/2014, 14:23

Ngày tải lên: 05/08/2014, 15:20

Ngày tải lên: 05/08/2014, 15:20

... lower bounds for the β j were actually equalities, then the lower bound given by (6) is sharp. We will accomplish this by showing that the M n (T), the number of zeros of ξ(s) with multiplicity ... Let ρ (j) = β + iγ denote a zero of the j th derivative ξ (j) (s), and denote its multiplicity by m(γ). Define the following counting functions: N (j) (T )=  ρ (j) =β+iγ 1 zeros of ξ (j) (σ +...

Ngày tải lên: 07/08/2014, 06:20