Báo cáo toán học: "Minimally Intersecting Set Partitions of Type B" potx

... tudy of minimally intersecting set partitions, we investi- gate minimally intersecting set partitions of type B. Our main result is a formula for the number of m inimally intersecting r-tuples of ... structure of the lattice of type B n parti- tions of the set {±1, ±2, . . . , ±n}. The lattice of type B n set partitions has been studied by Reiner [8]....

Ngày tải lên: 08/08/2014, 11:20

... journal of combinatorics 18 (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 ... vertices with the vertex set (S \ X) ∪ (T \ Y ). The vertex set of one component of the bipartite graph consists of one block of a partition of S \ X and o...

Ngày tải lên: 08/08/2014, 12:23

... counting the number of excedances of an ordinary cycle. This implies that as the type A case, the number of excedances of type B in π equals to the total number of excedances of type B in a ll prime (resp. ... extension of the decomposition of derangements given by Kim and Zeng [11] in their combinatorial proof of (1.3). Combinatorial Proof of Theorem 3.3. First, we...

Ngày tải lên: 08/08/2014, 01:20

... numb er of singleton blocks equal to {1} in partitions of [n] is equal to the number of partitions of [n − 1] we see that the generating function for the number of singleton blocks in all partitions ... differentiation of exponential generating functions only means a shift of coeffi- cients, we find that the total of sumrec, summed over all set partitions of n into k s...

Ngày tải lên: 08/08/2014, 12:22

... every 4-subset of A 2 has 6 2subsets that belong W (b k ,b l ). If Card(A 2 ) < 4, Card(A 1 ) ≥ 3, and every 3-subset of A 1 has 3 2- subset belong W(b k ,b l ). Let A  1 be such a subset. Then ... Z(5,z)+2η ≤ cr D 5,z . Proof:Let h(t k ) be the sum of the number of crossings of the edges of t k with edges not incident on any vertex of T .Lett min be the element of T such...

Ngày tải lên: 07/08/2014, 07:21

... context of Pieri’s rule for symmetric functions, our construction is more related to that of weights of arcs, arising in the context of ideal triangulations of marked Riemannian surfaces [13] Proof. ... denotes the number of partitions of a into ℓ blocks, {a 1 , a 2 , . . . , a ℓ }, such that ∀i ≤ ℓ, a i ≥ σ. the electronic journal of combinatorics 17 (2010), #R175 10 Proof....

Ngày tải lên: 08/08/2014, 12:23

... minimum dominating set of G, one may ignore V 1 and only consider dominating sets that extend V 2 . Proof. For any dominating set V 1 ∪ V 3 of G, V 2 ∪ V 3 is a dominating set. Hence, as |V 1 | ... the features of dynamic programming [8, Chapter 15]. During the search, we maintain sets of dominated (rather than dominating) vertices. On each level of the BFS, we have a collectio...

Ngày tải lên: 08/08/2014, 14:23

... the largest partitions into 4 and 3 sets; further evidence that 160 may be maximal for 5 set partitions. For partitions of 44 into 4 sets all 24 = φ(45) symmetric sum-free partitions of the 273 ... partitions are equivalent to partitions with large depth which can be build by first finding partitions in fewer sets. The partitions P of k = 6 and 7 sets given at the end of...

Ngày tải lên: 07/08/2014, 06:20

... of the colorings of the edges of S and reverses the colors in the other 14. Figure 6 shows the effect of H S on S for a complete (up to rotations) set of colorings of S. From the perspective of ... entries of a height function matrix for an ASM should be thought of as ofset from the entries of the ASM itself. If the entries of the ASM lie on vertices in the square grid,...

Ngày tải lên: 07/08/2014, 06:20

... column-index of the first and the last 1-cell in the i-th row of M. We will write f i and  i instead of f i (M) and  i (M) if there is no risk of confusion. Before we formulate the proof of Lemma ... from Theorem 42. 4 The patterns of size four 4.1 Enumeration of 1123-avoiding partitions To complete the classification of the equivalence classes of the patterns of length...

Ngày tải lên: 07/08/2014, 15:23