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Further Investigations Involving Rook Polynomials With Only Real.. - Haglund (1998) (2 citations) (Correct)
....a 23 a 45 ) 1 2 3 a 13 12 a 23 a 4 5 a 45 Figure 1. A weighted graph. In sections 2 and 3 we discuss two other factorization theorems. One involves generalized rook numbers which count cycles in digraphs, which were studied by Chung and Graham and also by Dworkin and Gessel [ChGr] Dwo] Ges1] [Ges2]. Another is a recent result of Reiner and White [ReWh] dealing with the matching numbers of graphs of shifted Ferrers shape. In both cases we show how the underlying factorization theorem motivates a conjecture involving arrays of real numbers which are weakly increasing in two directions. We ....
.... (1; A) 0: On the other hand, after placing a rook on the corner of the rectangular region of A where the border segments of lengths h p and d p meet, the unattacked portion of A remaining has shape A p , which shows r Max(Ap) 1 (1; A) 6= 0: We will require the following result of Gessel [Ges2], which holds for any admissible board A; T (y; y; z; A) n rooks on A # of rooks on nonzero squares of A : 9) The y = 1 case of Theorem 2 below, without the interlacing hypothesis, reduces to the a ij 2 f0; 1g case of the MCP Conjecture. Theorem 2. Let A be the Ferrers board of Fig. 5; ....
I. M. Gessel, private communication.
Deformations of Coxeter Hyperplane Arrangements - Postnikov, Stanley (1997) (14 citations) (Correct)
....exponential generating function for the sequence f n . A plane binary tree B on the vertices 1; 2; n is called a local binary search tree if for any vertex i in T the left child of i is less than i and the right child of i is greater than i. These trees were first considered by Ira Gessel [11]. Let g n denote the number of local binary search trees on the vertices 1; 2; n. By convention, g 0 = 1. r r r r r r r r r r Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma ....
I. Gessel, private communication.
Alexander Postnikov - Department Of Mathematics (Correct)
....interpretations of the numbers F n 4.1 Local binary search trees. A local binary search (LBS) tree is a labeled binary tree such that every left child has a smaller label than its parent, and every right child has a larger label than its parent. LBS trees were first considered by Ira Gessel [2]. Theorem. For n 1 the number of LBS trees on the set [n Gamma 1] is equal to the number F n of intransitive trees on the set [n] Proof. Let I n be the set of intransitive trees on [n] with a chosen root. Let B n be the set of LBS trees on [n] such that the root has only one child (left or ....
I. Gessel, private communication.
Enumeration in Algebra and Geometry - Postnikov (1997) (3 citations) (Correct)
....: 1.4.2 Local binary search trees A plane binary tree on the vertices 1; 2; n is called a local binary search tree (LBS tree, for short) if for any vertex i the left child of i is less than i and the right child of i is greater than i. These trees were first considered by Ira Gessel [21], and were studied in [42] The name local binary search tree was suggested by Richard Stanley [53] see also [44] r r r r r r r r r r Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma ....
I. Gessel, private communication.
Further Investigations Involving Rook Polynomials With Only Real.. - Haglund (1998) (2 citations) (Correct)
....of an array, and is related to the matching polynomial in the same way that the MCP conjecture is related to the rook polynomial. Another involves generalized rook numbers which count cycles in digraphs, as studied by Chung and Graham, Chow, Dworkin, and Gessel [ChGr] Cho] Dwo] Ges1] [Ges2]. Section 3 contains various technical lemmas which we utilize in Sections 4 and 5 to prove the analogs of Theorem 1:1 as well as other special cases of these conjectures. In particular, we show (Theorem 2:2) that for certain threshold graphs a stronger result than the Heilmann Lieb Theorem holds. ....
.... hand, after placing a rook on the corner of the rectangular region of A where the border segments of lengths h p and d p meet, the unattacked portion of A remaining has shape A p , which shows r Max(Ap) 1 (1; A) 6= 0: In order to prove Theorem 5:4, we also require the following result of Gessel [Ges2] (see also [Hag2, eq. 21) which holds for any admissible board A; T (y; y; z; A) X C n rooks on A z # of rooks on nonzero squares of A y cyc(C) 11) Theorem 5:4. Let A be the Ferrers board of Fig. 5; assume y 0 and A is admissible. Then as a polynomial in z, T (y; y; z; A) has ....
I. M. Gessel, private communication.
Deformations of Coxeter Hyperplane Arrangements - Postnikov, Stanley (1997) (14 citations) (Correct)
....exponential generating function for the sequence f n . A plane binary tree B on the vertices 1; 2; n is called a local binary search tree if for any vertex i in T the left child of i is less than i and the right child of i is greater than i. These trees were first considered by Ira Gessel [11]. Let g n denote the number of local binary search trees on the vertices 1; 2; n. By convention, g 0 = 1. r r r r r r r r r r Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma ....
I. Gessel, private communication.
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