Báo cáo toán hoc:" q-Counting descent pairs with prescribed tops and bottoms"

Báo cáo toán hoc:" q-Counting descent pairs with prescribed tops and bottoms" docx

... invXc(c), and each of the +’s to the left of xicontributes 1 to rlmajX,Y(c). Thus itthe electronic journal of combinatorics 16 (2009), #R111 12 q-Counting descent pairs with prescribed tops and ... Series, in “Encyclopedia of Math. and its Applications,” Cambridge Univ. Press, Cambridge, MA, 1990 .[4] J. Hall, J. Remmel, Counting Descents with Prescribed Tops and Bottoms,math.CO/0610608[5] ... 05A15AbstractGiven sets X and Y of positive integers and a permutation σ = σ1σ2· · · σn∈ Sn,an (X, Y ) -descent of σ is a descent pair σi> σi+1whose “top” σiis in X and whose“bottom”...

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Báo cáo toán học: "On the functions with values in [α(G), χ(G)]" docx

... V \M, if verticesi and j are non-adjacent and el· f(i) =0,el· f(j) = 0 (hence, l is adjacent to i and j),then a vertex m ∈ M (m = l) exists such that em· f(i) =0andem· f(j) = 0 ... ei∈ Rkto vertex ai∈ A and vector sjto vertex bj∈ B for i =1, ,k, j =1, ,n.To form edge set E(G):• join every vertex from A with every vertex from B;• join bi and bjfrom B iﬀ si· ... from{e1,e2,e3}.Lemmas 1 and 2 imply the following theorem.Theorem 1 There exists a graph G such that α(G)=d(G) and α(G) <χ(G).Corollary 1 There exists graphs G such that α(G)=θ(G)=r(G) and α(G) <χ(G).The...

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Báo cáo toán học: "Filling a box with translates of two bricks" docx

... rectangles (bricks), typeA with dimensions (a1, ,ad) and type B with dimensions (b1, ,bd). We want to usetranslates of such bricks to ﬁll completely, and with no overlaps (except at ... part of the zero sets of the ﬁrst and second terms in theleft hand side of (2) respectively. But the intersection of these hyperplanes, on which setthe left hand side vanishes, contains points ... brick A can tile Q alone, and there is nothing to prove. So we may assume that one of them is not an integer, saya−11/∈ Z.Bychoosingi =1andj =2, 3, ,d in turn, and using our key observationabove,...