Báo cáo toán học: "A Combinatorial Interpretation of the Area of Schr¨der Paths o" pps

... bijective proof for this recurrence in terms of the area of Schr¨oder paths. In 1976 Kreweras [7] proved that the sum of the areas of the regions lying under the elevated Schr¨oder paths satisfies ... satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of...

Ngày tải lên: 07/08/2014, 06:20

... to the product of the weights of each of its entries, here the weight of entry s in T is taken to be the weight of the corner square containing s in the pa rt itio n obtained from the shape of ... is the the number of columns of height equal to the height of the column of i in the tableau obtained by removing all the entries j > i from T , and b...

Ngày tải lên: 08/08/2014, 12:22

... +32t − 1. Then f(n) ≥ n +32t − 1, for n ≥ n t . This completes the proof of the theorem. From the above theorem, we have lim inf n→∞ f(n) − n √ n ≥  2+ 2562 6911 , which is better than the previous ... denote the number of vertices, and (G) denote the number of edges. In this paper, we construct a graph G having no two cycles with the same length which leads to the fol...

Ngày tải lên: 07/08/2014, 06:22

... s 2 n . (1) where b(i) is the number of times the integer i occurs in the sequence T . Note that the sum of all b(i), 1 ≤ i ≤ n,isequaltol, the length of T . 3 A simple proof for the existence of exponentially balanced ... u R 1 ,n+1,u 1 ,n+2,u R 1 ,j 0 , u R 0 ,n+2,u 0 ,n+1,u R 0 ,n+2, is the transition sequence of an (n +2)-bit Gray code. In the proof of Theorem...

Ngày tải lên: 07/08/2014, 08:22

... agree above row r and to the left of column c (and the definitions of r and c) we see that the only possibility for λ/µ is (r, c), completing the proof of this case and the theorem. Acknowledgements. ... ). Finally, recall that the conjugate of a partition λ is the partition λ  obtained by flipping the diagram of λ across the NW-SE axis; it follows that λ  i counts...

Ngày tải lên: 07/08/2014, 15:22

... : the uppermost cell of the rightmost column of the polyomino has the maximal ordinate among all the cells of the polyomino; U2 : the lowest cell of the rightmost column of the polyomino has the ... (k) g (k) g . The root of the tree is (1) b , which is the label of the one-cell polyomino. the electronic journal of combinatorics 14 (2007), #R57 11...

Ngày tải lên: 07/08/2014, 15:22

... study the asymptotic behaviour of LA n (a). As a matter of notation, we will use generically, the expression LA n (µ) for the distribution of the length of the longest alternating subsequence of ... s m . the electronic journa l of combinatorics 17 (2010), #R168 10 Theorem 4.2 Let LA n (x 0 , . . . , x n ) be the length of the longest alternating subsequence of...

Ngày tải lên: 08/08/2014, 12:23

... exactly once. In this paper, we give a short proof for the result of Noonan [3] that the number of permutations of length n containing exactly one occurence of pattern 321 is 3 n  2n n−3  . (To be ... π 1 cπ 2 bπ 3 aπ 4 . We will replace each of the points (j, c) and (ℓ, a) on the permutation diagram of π with two points as follows. Let 0 < ǫ < 1 and let ˆπ be a pe...

Ngày tải lên: 08/08/2014, 14:23

... 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

... 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 follow...

Ngày tải lên: 07/08/2014, 08:22