Báo cáo toán học: "DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES" pps

... X ∗ and n ≥ 1 such that α = , x = αβ, y = βα and u = (αβ) n α. Lemma 5.2. Let x, u, v ∈ X + .Ifu = xv = vy and |v|≥|u|/2,thenu/∈ HQ. Proof. By Lemma 5.1, there exist α, β ∈ X ∗ with αβ =  and ... be an alphabet with |X|≥2.Ifaw /∈ HQ and wb /∈ HQ where w ∈ X ∗ and a, b ∈ X,thenawb /∈ HQ. Proof. Assume that aw /∈ HQ and wb /∈ HQ.Thenaw ∈ v ⊗ v ⊗···⊗v and wb ∈ u ⊗ u ⊗ ···⊗ u...

Ngày tải lên: 06/08/2014, 05:20

... by Moon and Moser [3]. Furthermore, I(n, n − 2) = 2 n−2 − 2 and the extremal graph (unique up to isomorphism) coincides with K 2,n−2 for ... by n − k cliques, then ω(G) ≤ k +1 if k =2or 3 and ω(G) ≤  k k/2  if k ≥ 4. Moreover, these bounds are sharp for every n ≥ k +1if k =2or 3 and n ≥ k +  k k/2  if k ≥ 4. Proof:LetC = {c 1 , ... c i \  j=i c j for every i =1, ,n− k.Denoti...

Ngày tải lên: 07/08/2014, 06:23

... non-negative integer and α, β, γ non-negative integers with α + β + γ = n + 2. Then the number of triangulations o f the rooted polygon P → n+2 with α vertices of colour a, β vertices of colour b, and ... 1. Moreover, upon setting k = 0 and β 0 = d + 1 in Theorem 2.2, we obta in Formula (1.2) (and (1.1) in the further special case where d = 2). 3 Generating functions and the Lagra...

Ngày tải lên: 08/08/2014, 14:23

... with equations (10) and (11) respectively. For (iv) and (v), use Theorem 5.1 together with equations (13) and (14). DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES Michael ... polynomials at √ 3, generalized Euler and class numbers, and 3-signed permutations. In [17] Shanks defined positive integers c a,n (for integer a ≥ 1andn≥0) and d a...

Ngày tải lên: 07/08/2014, 06:20

... 1973. 6. V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in Sl(n)/B, Proc. Indian Acad. Sci. 100 (1990), no. 1, 45–52. 7. L. Lorentzen and H. Waadeland, Continued fractions ... PERMUTATIONS, CONTINUED FRACTIONS, AND CHEBYSHEV POLYNOMIALS Toufik Mansour ∗ and Alek Vainshtein † ∗ Department of Mathematics † Department of Mathematics and Department of Compu...

Ngày tải lên: 07/08/2014, 06:20

... #R5 14 Billey and Warrington [BW] introduced the 321-4-gon-avoiding (or 321-hexagon-avo- iding) permutations as a class in S n whose Kazhdan-Lusztig and Poincar´e polynomials, and the singular ... Mansour and A. Vainshtein, Layered restrictions and Chebychev polynomials (2000), Annals of Combinatorics 5 (2001) 451–458. [MV4] T. Mansour and A. Vainshtein, Restricted permutations...

Ngày tải lên: 07/08/2014, 07:21

... techniques of Bandlow and Killpatrick [2] and Bandlow, Egge, and Killpatrick [1] to compute this bijection directly. To define our bijection, we first need to introduce some notation. For all n ≥ 0and all ... enumerations and generating functions for permutations which avoid 1243, 2143, and certain additional patterns. We also give generating functions for permutations which avoid 12...

Ngày tải lên: 07/08/2014, 07:21

... between Random Matrix Theory and Robinson-Schensted-Knuth type problems discovered recently by Gessel [3], Baik, Deift and Johansson [4] and Rains [6] (see also Aldous and Diaconis [5] and the ... consequences of the Keating and Snaith conjecture The above results lead to an interesting interpretation of the Keating and Snaith conjec- ture. Once m is a non-negative integer, the...

Ngày tải lên: 07/08/2014, 07:21

... hyperplane x 0 =1in R N+1 and denote by C P theconeinR N+1 generated by ˜ P and by E P the semigroup of integer points in C P .LetL be the linear span of C P in R N+1 and A be a set of integer linear forms ... suitable positive integer multiple we find an integer point α in L satisfying f(α) < 0andg(α) > 0 for all g ∈Aother than f. Letting a = f(α), we may choose a nonnegat...

Ngày tải lên: 07/08/2014, 08:22

... algorithmically hard in this case. On the other hand, Lov´asz [5] showed that if no two consecutive integers in f(v) differ by more than two, then there is a necessary and sufficient condition for an f-factor. ... Concluding Remarks We mention a possible direction for extending the results of Theorem 1 and Theorem 2. Alon, Friedland and Kalai [3] proved that for any prime power p, every gr...

Ngày tải lên: 07/08/2014, 15:22