Báo cáo toán học: "A Concise Proof of the Littlewood-Richardson Rule" pdf

... extension of the Littlewood-Richardson rule [Z]. Taking the specialization λ = , we obtain the decomposition of s µ/ν into Schur functions; it is simpler than the traditional formulation of the Littlewood-Richardson rule ... b i free k + 1’s in row i of T , then there should be b i free k’s and a i free k + 1’s in row i of σ k (T ). In the following, T ≥j denotes th...

Ngày tải lên: 07/08/2014, 06:23

Ngày tải lên: 05/08/2014, 09:46

... ···(x−n+1). Then we have the following reciprocity theorem. Theorem. For any board B ⊂ [d] × [d], R(B; x)=(−1) d R(B; − x − 1). The existing proofs derive this as a corollary of other reciprocity theorems, ... subset of [d] × [d](where[d]isdefinedtobe{1, 2, ,d})andtherook numbers r B k of a board are the number of subsets of B of size k such that no two elements have the sa...

Ngày tải lên: 07/08/2014, 06:20

... number of boxes in the first row is equal to the size of the first part, the number of boxes in the second row is equal to the size of the second part, etc. For example, the Ferrers diagram for the ... that the set S consists of pairs of partitions with no primed parts. In the definition of φ −1 when there is a primed integer in the image, we consider only...

Ngày tải lên: 07/08/2014, 08:20

... of the hive give λ • the differences on the Northeast side of the hive give µ • the differences on the South side of the hive give ν where all differences are computed left-to-right throughout the ... vertex. The constant coordinate assigned is then the label on ∆ counterclockwise of the extra vertex, minus the label on ∆ clockwise of the extra vertex. The v...

Ngày tải lên: 07/08/2014, 08:20

... me to the area of log-concave sequences. Abstract We provide a combinatorial proof for the fact that for any fixed n, the sequence {i(n, k )} 0≤k ≤ ( n 2 ) of the numbers of permutations of length ... i(p) of inversions, and the study of numbers i(n, k)ofn-permutations having k inversions, is a classic area of combinatorics. The best-known result is the following...

Ngày tải lên: 07/08/2014, 08:22

... 1986. the electronic journal of combinatorics 12 (2005), #R18 8 Theorem 2.3 (Aztec diamond theorem) The number of domino tilings of the Aztec diamond of order n is 2 n(n+1)/2 . Remark: The proof of ... Kirkland [2] give a proof by considering a matrix of order n(n +1) the determinant of which gives a n . Their proof is reduced to the computation of the dete...

Ngày tải lên: 07/08/2014, 08:22

... [X] 4 exists. In the light of Theorems 2 and 3, a positive resolution of the following question would be the best improvement of Theorem 1 possible. Question 1. If X is an order with uncountable character, then ... short proof The following theorem first appeared in [3, Theorem 31, pp. 447–457]. (Actually, this is not quite true. The theorem there was claimed only for uncount...

Ngày tải lên: 07/08/2014, 06:20

... 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 ... [5]. Zeilberger gave a q-Foata proof of the q-Pfaff-Saalsch¨utz identity [20], inspired by Foata’s bijective proof of the Pfaff-Saalsch¨utz identity [6]. In view...

Ngày tải lên: 07/08/2014, 06:22

... transpose two of the rows or two of the columns in the corresponding diagram. One must be careful, however, to preserve row and column sums. If two of the row sums are the same, or if two of the column ... 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 interest...

Ngày tải lên: 07/08/2014, 08:20