Báo cáo toán học: "Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules" docx

... 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) ... t). In other words, the statistics for labelled paths inside the triangle with vertices (0, 0), (0,n), and (mn, n) give a combinatorial interpretation for the Hilber...

Ngày tải lên: 07/08/2014, 08:20

... 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

... either two A v ’s of the form {x, z} and {y,w} or, of the form {x, w} and {y, z}. But this contradicts our assumption that the two element A v ’s are never disjoint. This completes the proof ... contains the value of γ(n, δ) for n ≤ 16, 0 ≤ δ ≤ n − 1 if the value is known, otherwise upper and lower bounds for γ(n, δ). We establish these values in Sections 2, 3 and 4. TIG...

Ngày tải lên: 07/08/2014, 06:20

... definition of δ k (n). The theorem therefore follows. In the next section we show that if k = 3 then the above theorem holds for n ≥ 6. Note that the results in this section are stated for fixed ... a i , 7. 0 the electronic journal of combinatorics 7 (2000), #R58 15 There are at most  s 2  elements for each of the types (1), (2), (3) and (4). There are at most s eleme...

Ngày tải lên: 07/08/2014, 06:22

... Partition of the vertex set In the proof of the conjecture we will use two special sets, obtained by an operation of completion of a certain set to a set that dominates a specified set of vertices of ... neighbors of any vertex of D  G .Set D  G = D G \ D  G denoting the vertices of D G which do not have private neighbors (so they must enjoy condition (i) of th...

Ngày tải lên: 07/08/2014, 08:22

... Yet another reformu- lation shows that the cycle index series is also  T ∈T Z(Aut(T )), where Z(Aut(T )) is the cycle index of the automorphism group of T . The following formula for the cycle ... Labelle’s) for this formula, for which only the classical single-variable form of Lagrange inversion will be necessary; then, some immediate corrolaries are stated. Finall...

Ngày tải lên: 07/08/2014, 13:21

... 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

... orders, as well as for coding of chemical structures [12, 13, 26, 32]. Therefore, the ordering of molecular graphs in terms of their Hosoya indices is of int erest in chemical thermodynamics. Let ... This completes the proof of (i). By a similar discussion as in the proof of (i), we may also show that (ii)-(iv) are true. We omit the procedure here. This completes the p...

Ngày tải lên: 08/08/2014, 12:22

... 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, ... + b = v. Then a = xv = x(a + b) = x 2 v + xyv = x 2 v = xa. Then a is in the 1-space of x, and, simlarly, b is in the 1-space of y. Then the 1-space...

Ngày tải lên: 08/08/2014, 12:23

... the opposite of the loga- rithm of the normalised sum of probabili- ties of all possible word associations defined by the edge (for the edge between multiple (el) access (e2) to the French ... the cost of an edge involving an empty word to be e, an arbitrary small positive value, and the cost of all the other edges (i.e. the edges from SoP and SiP) to...

Ngày tải lên: 17/03/2014, 07:20