Báo cáo toán học: "A Combinatorial Formula for the Hilbert Series of bigraded Sn-modules" ppsx

... t). The basic idea of the proof of Theorem 3.1 is to show F µ (q, t) and G µ (q, t) satisfy the same recursion in the hook case. Proof. We first note the Garsia-Haiman recursion for the Hilbert series ... 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...

Ngày tải lên: 08/08/2014, 12:22

... invalid in the other.) This lemma, together with the discussion preceding it, completes the proof of formula (18). We leave to the reader the task of showing that this formula reduces to formula (4) ... study the combinatorial properties of these statistics, obtaining an explicit summation formula for their generating function and giving bijective proofs of th...

Ngày tải lên: 07/08/2014, 08:20

... algebras of Coxeter g roups. Section 2 establishes notation and describes the relevant background necessary for the rest of the paper. For further background information on the properties of the symmetric group, ... 4 of the demipotents. These diagram demipotents are + + − + ++, + − + − ++, and their siblings. An open problem is to find a formula for the nilpotenc...

Ngày tải lên: 08/08/2014, 12:23

... : 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

... 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

... inv(σ) denote, resp ectively, the descent number and inversion number of σ in the symmetric group S n of degree n. We give an algebraic proof and a combinatorial proof of the recurrence relation. 1 ... :=  σ∈S n t des(σ) . 3 A combinatorial proof We give a combinatorial proof of Theorem 2.2 in the present section. Recall that elements of S n+1 can be obtained by inser...

Ngày tải lên: 08/08/2014, 11:20

Ngày tải lên: 05/08/2014, 15:20

... well-defined. The choices for the initial position of F in each iteration might effect the outcome of the algorithm. But, it has been shown (see Section 3.9 of [S] for a proof) that for all choices of the F’s, ... algorithm works for any choice of σ. Theorem 2.3. The above algorithm for determining m i (T ) works for all choices of the ordering σ of the...

Ngày tải lên: 07/08/2014, 06:20

... (4) in [22]. 2 In the derivation, the value of k appears in relation to the geometry of FPLs, and the exact range of k for which the formula is pr oved is not made clear. In the present work we ... that the formula be true for one value of k – the explicit value being irrelevant since the result should be independent of k. See also Theorem 4.2 of [4] (whi...

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