Báo cáo toán học: "MacWilliams identities and matroid polynomials" doc

... the identities (3.1), (3.2), (3.3), and (3.4) show that A [m] C   {z e } E   depends only on the matroid M C  . Hence, condition 2 in Theorem 15 is satisfied by the identities (3.2) and (3.3), ... several MacWilliams type identities, including those by Kløve and Shiromoto, and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid...

Ngày tải lên: 07/08/2014, 06:23

... generalizes a determinant identity of Andrews and Burge [2]. Finally, in Section 6 we translate our determinant identities of Sections 4 and 5 into constant term identities which seem to be new. Auxiliary ... (4.19) vanishes if and only if i ≤ 2v − m and i + j>2v−m. (4.27) The second term in (4.19) vanishes if and only if  v − 1 2  ≤ i ≤ v + j − 2 2 . (4.28) The third term...

Ngày tải lên: 07/08/2014, 06:20

Ngày tải lên: 05/08/2014, 15:20

... [37], Multivision [26], partizan octal and subtraction games [38] and extensions of Conway’s ‘short games’ [52]. He has been especially interested in annihilation games [10, 53, 54, 56] and in games using Cedric ... workshops, conferences and seminars – he displayed 11 different games at the 1986 Strens Memorial Conference in Calgary [21], and was on the organizing committee of both...

Ngày tải lên: 07/08/2014, 06:23

... n.LetI B and I S be subsets of [k] and let B = {C i : i ∈ I B } and S = {C i : i ∈ I S }. We refer to the sets B and S as the bases and summits of G respectively. A set C i ∈BiscalledabaseofG and ... line, and then we connect n to the lefthand endpoint of the first cycle, the righthand endpoint of each cycle to the lefthand endpoint of the cycle following it, and we connect the...

Ngày tải lên: 07/08/2014, 06:23

... follows from Theorem 2 and item 4 above. ✷ Corollary 1 There exists a partition of n with BG-rank = j if and only if j + n is even and j(2j − 1) ≤ n. 4 Congruences Berkovich and Garvan’s motivation ... combinatorics 13 (2006), #N18 2 References [1] Alexander Berkovich and Frank Garvan, On the Andrews-Stanley refinement of Ra- manujan’s partition congruence modulo 5 and generalizati...

Ngày tải lên: 07/08/2014, 13:21

... 0) and (i, 1), and between (i, b) and (i, b+1), for each i ∈ [a], and vertically between (0, j) and (1, j), and between (a, j) and (a+1, j), for each j ∈ [b], and to include in each path P k the ... Lexicographic and Antilexicographic Ordering Pairs In this section, the lexicographic and antilexicographic ordering pairs ( ˆ L, <) and ( ˇ L, >), with ˆ L and ˇ L d...

Ngày tải lên: 07/08/2014, 15:22

... (3.12), (3.13),(3.14) and (3.15) and equating the results pairwise as follows. Equations (3.8) and (3.9) give (1); equations (3.10) a nd (3.11) give (2); equations (3.12) and (3 .1 3) give (3); and equations ... math and state and prove the following theorem. Theorem 7.2. T he unique sol ution to the recurrence equation (7.2) with m −1 and n ∈ Z and with boundary given by f...

Ngày tải lên: 08/08/2014, 12:22

... of two 2-matroids Let Q 1 and Q 2 be 2-matroids. When Q 1 and Q 2 have the same ground-set and skew classes we shall use Q 1 + Q 2 to denote the parity system which is the sum of Q 1 and Q 2 . ... Q 1 and Q 2 . When Q 1 and Q 2 have disjoint ground-sets we shall use Q 1 ⊕ Q 2 to denote the 2 -matroid which is the direct sum of Q 1 and Q 2 . Let Q = (U, Ω, r) be a 2 -matroid a...

Ngày tải lên: 07/08/2014, 06:20

... collection of forests of height at most m,andletp m (n)anda m (n)beits counting functions. For m = 0 one has p 0 (1) = 1, and otherwise p 0 (n) = 0; and a 0 (n)=1 for all n.Form = 1 clearly p 1 (n) ... F m gives a forest in F m−1 , and indeed this operation is a bijection between the trees of F m and all of F m−1 . Thus for m ≥ 1 p m (n)=a m−1 (n − 1) . By Theorem B and induction on...

Ngày tải lên: 07/08/2014, 08:20