Báo cáo toán học: "A recurrence relation for the “inv” analogue of q-Eulerian polynomials" ppsx

... A recurrence relation for the “inv” analogue of q-Eulerian polynomials Chak-On Chow Department of Mathematics and Information Technology, Hong Kong Institute of Education, 10 Lo ... relation algebraically. In the final section, we give a combinatorial proof of this recurrence relation. 2 The recurrence relation We derive in the present section the rec...

Ngày tải lên: 08/08/2014, 11:20

... 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 f...

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 ... code the transition counts of which are all the same power of two, or are two successive powers of two. A proof for the existence of e...

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. ... k-deletion of µ can contain the cell (k, k + 1) and that no k-deletion of λ can contain the cell (k + 1, k). Therefore every k-deletion of µ and of λ is actually...

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 ... interpretation of the closed formula (10) for the number of con- vex permutominoes. We remark that the expression of f n resembles the expression (1)...

Ngày tải lên: 07/08/2014, 15:22

... presentation of the fundamental group of the knot complement. For a square matrix M with entries in CG the trace tr CG (M) denotes the coefficient of the unit element in the sum of the diagonal elements. The ... k −2 AA ∗ ) n  (4) The value of k can be chosen to be the product of (g −1) 2 and the maximum of the 1-norm of the entries in A (see [L¨uc94...

Ngày tải lên: 07/08/2014, 15:23

... be the number of cells (including the possible circle at the end of the row) to the right of s. Let a lso ℓ Λ (s) be the number of cells (not including the possible circle at the bottom of the ... 1))  . (9) Proof. The proof will proceed by cancellation of certain terms in the L.H.S. of the equation to obtain the R.H.S. Figure 3 illustrates the g ene...

Ngày tải lên: 07/08/2014, 21:21

... follows for (π, f ) ∈ S n,A : If R (π,f) = ∅, let φ((π, f)) = (π, f). Otherwise, let (i, j) ∈ R (π,f) be minimal under the lexicographic ordering of R (π,f) . Let ˜π = (i, j) ◦ π, the product of the ... of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original. the e...

Ngày tải lên: 08/08/2014, 01:20

... 41 λ a b f d ce Figure 42 Proof . We have the following cases: 1. Suppose that a comes from a fork. Since the degree of o is 4, o must be the white node in the fork. Thus, f is contained in the same fork. Then node ... consistent. 2. The second possible neighborhood in S o comes from a fork and a diamond, see Figure 13. Note that the direction of the fork and the diamond...

Ngày tải lên: 08/08/2014, 14:23