S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71--96, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to product forms and uniform generating trees. The idea of generating trees has surfaced occasionally in the literature. West introduced it in the context of enumeration of permutations with forbidden subsequences [27, 28] this idea has been further exploited in closely related problems [6, 5, 12, 13]. A major contribution in this area is due to Barcucci, Del Lungo, Pergola, and Pinzani [4, 3] who showed that a fairly large number of classical combinatorial structures can be described by generating trees. A form equivalent to generating trees is well worth noting at this stage. Consider the ....

S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71-96, 1998.

....: k) n occurrences) This corresponds to walks with multiplicities, or if one wants, to distinguish two occurrences of the same label in a succession rule by colouring them in two di erent colours. The method of generating trees was also successfully used by West [25] Dulucq, Gire, and Guibert [12 14], for the enumeration of permutations with forbidden sequences (see Fig. 2) In fact, the kind of rewriting rules under consideration here were intensively studied partly because they are useful to solve some cases of the following famous conjecture: Conjecture 1 (Stanley Wilf) For any given ....

S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71-96, 1998.

....third one, F (t; 1) 1 2 X n 1 3 n 1 (n 1) 2n 1) 3n n t n : There is no doubt that these simple formulae call for more combinatorial explanations. Such explanations have actually been provided, at least to a certain extent: see [11] for paths, 5] for constellations, and [10, 14] for two di erent recursive descriptions of bijections between two stack sortable permutations and non separable planar maps, which are known to be counted by (4) see [7] There is no doubt either that the functional equation approach has allowed us to prove these results, and that it is ....

S. Dulucq, S. Gire and O. Guibert, A combinatorial proof of J. West's conjecture. Discrete Math. 187 (1998) 71-96.

....= 12 : n, i.e. if it occurs in the last three columns of the sorting tree. West characterized these permutations in terms of forbidden patterns [22] and conjectured that their number is b n = 2(3n) 2n 1) n 1) This conjecture was rst proved by Zeilberger [23] Two bijective proofs [10, 13] were found later, based on the fact that b n is the number of non separable planar maps [5, 7] Note that the corresponding generating function P b n x n is cubic over IR(x) ffl Sorted permutations A permutation 2 Sn is sorted if it belongs to Pi(S n ) In other words, the sorted ....

....where Y = Y (x) is the formal power series in x dened by Y = 2xY (1 Gamma Y )C(x; Y ) xY (Y Gamma 1)C 0 (x) x(1 Gamma Y ) Computing the resultant of Delta and Delta= y gives the algebraic equation satised by C 0 (x) Remarks 1. The rst part of the above proposition was already proved in [10, 13, 23]. 2. Let c n denote the coeOEcient of x n in C 0 (x) The numbers c n have large prime factors (see Table 1) We can prove they are not hypergeometric as follows: we rst construct the linear recurrence with polynomial coeOEcients they satisfy (using, for instance, the Maple package Gfun [18] ....

S. Dulucq, S. Gire and O. Guibert, A combinatorial proof of J. West's conjecture, Discrete Math. 187 (1998) 7196.

....to product forms and uniform generating trees. The idea of generating trees has surfaced occasionally in the literature. West introduced it in the context of enumeration of permutations with forbidden subsequences [27, 28] this idea has been further exploited in closely related problems [6, 5, 12, 13]. A major contribution in this area is due to Barcucci, Del Lungo, Pergola, and Pinzani [4, 3] who showed that a fairly large number of classical combinatorial structures can be described by generating trees. A form equivalent to generating trees is well worth noting at this stage. Consider the ....

S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71--96, 1998.

....to product forms and uniform generating trees. The idea of generating trees has surfaced occasionally in the literature. West introduced it in the context of enumeration of permutations with forbidden subsequences [24, 25] this idea has been further exploited in closely related problems [4, 5, 10, 11]. A major contribution in this area is due to Barcucci, Del Lungo, Pergola, and Pinzani [3, 6] who showed that a fairly large number of classical combinatorial structures can be described by generating trees. A form equivalent to generating trees is well worth noting at this stage. Consider the ....

S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71-96, 1998.

....uniform generating trees. The idea of generating trees that we have just described has surfaced occasionally in the literature. West introduced it in the context of enumeration of permutations with forbidden subsequences [18, 19] this idea has been further exploited in closely related problems [3, 4, 9, 10]. A major contribution in this area is due to Barcucci, Del Lungo, Pergola, and Pinzani [2, 5] who systematized the method under the name of ECO systems (ECO stands for Enumerating Combinatorial Objects ) while showing that a fairly large number of classical combinatorial structures are amenable ....

S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71--96, 1998.

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S. Dulucq, S. Gire, and O. Guibert. A combinatorial proof of J. West's conjecture. Discrete Mathematics, 187(1-3):71--96, 1998.

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