R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), 261--279.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow [1] we combine our involution with a result of Gasharov [5] to give a combinatorial proof of a special case of the (3 1) free Conjecture of Stanley and Stembridge [14] 1. Introduction We rst recall some de nitions from the theory of Young tableaux. Further exposition can be found in the texts of Fulton [3] Macdonald [9] Sagan [10] and Stanley [13] Let = 1 ; 2 ; l ) be a partition of the nonnegative integer n, denoted n, so is a ....

.... Gessel Viennot [6, 7] as well as the rim hook Robinson Schensted algorithm of White [16] and Stanton White [15] In section 3 we follow a suggestion of Chow [1] and combine our involution with a result of Gasharov [5] to prove a special case of the (3 1) free Conjecture of Stanley and Stembridge [14]. Finally, we end with a discussion of further work which needs to be done. 2. The basic involution First note that by Theorem 1.1, K (1 ) is just the number of special rim hook tableaux of shape where all hooks have size one (since such tableaux have sign 1) But since they must also ....

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R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), 261-279.

....by induction on z. Then, we consider the limit with respect to the filtration topology of Q, X1,X2, Let us note that for q = 0 we obtain the well known identity exp u n qn , while for q = 1 we obtain the identity (6.2. 6) The latter appears in a slightly different form in [49], Proposition 2.2, and is attributed to I. Gessel; hence Corollary 6.2.5 represents the q analogue of (6.2.6) Other types of combinatorial identities, not necessarily involving symmetric functions, can be derived from Corollary 6.2.4. For instance, let us consider the formal group law X) over the ....

R. Stanley and J. Stembridge. On immanants of Jacobi-Trudi matrices and permutations with restricted position. d. Combi. Theor'i/Ser. A, 62:261 279, 1993.

....[n] q n u 1 q [n Gamma 1] q n u Let us note that for q = 0 we obtain the well known identity exp Psi n S n u while for q = 1 we obtain the identity n0 Psi n u n u (n Gamma 1) n u : 3. 9) The latter appears in a slightly different form in [16], Proposition 2.2, and is attributed to I. Gessel; hence Corollary 3.8 represents the q analogue of (3.9) Other types of combinatorial identities, not necessarily involving symmetric functions, can be derived from Corollary 3.7. For instance, let us consider the ring H and the umbra b. We view ....

R. Stanley and J. Stembridge. On immanants of Jacobi-Trudi matrices and permutations with restricted position. J. Combin. Theory Ser. A, 62:261--279, 1993. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, U.S.A. E-mail address: lenart@math.mit.edu

....a version of this Law for XG itself in the category of vertex weighted graphs. This permits the derivation of a number of Stanley s results by straightforward induction. It is also possible to use this inductive approach to make progress on the (3 1) free Conjecture of Stanley and Stembridge [31]. Call a poset (partially ordered set) a b) free if it has no induced subposet isomorphic to the dijoint union of an a element chain and a b element chain. Any poset P has a corresponding incomparability graph, G(P ) whose vertices are the elements of P with an edge between two vertices if they ....

R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), 261--279.

....does have such a law and specializes to XG when the variables are allowed to commute. This permits us to further generalize some of Stanley s theorems and prove them in a uniform and straightforward manner. Furthermore, we make some progress on the (3 1) free Conjecture of Stanley and Stembridge [14]. 1 Introduction Let G be a finite graph with verticies V = V (G) fv 1 ; v 2 ; v d g and edge set E = E(G) We permit our graphs to have loops and multiple edges. Let XG (n) be the chromatic polynomial of G, i.e. the number of proper colorings : V f1; 2; ng. Proper means ....

....[5] In Section 3 we define YG and derive some of its basic properties, including the Deletion Contraction Law. Connections with acyclic orientations are explored in Section 4. The next three sections are devoted to making some progress on the (3 1) 1 free Conjecture of Stanley and Stembridge [14]. Finally we end with some comments and open questions. 2 Symmetric functions in noncommuting variables Our noncommutative symmetric functions will be indexed by elements of the partition lattice. We let Pi d denote the lattice of set partitions of f1; 2; dg : d] ordered by ....

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R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory (A) 62 (1993), 261-- 279.

....does have such a law and specializes to XG when the variables are allowed to commute. This permits us to further generalize some of Stanley s theorems and prove them in a uniform and straightforward manner. Furthermore, we make some progress on the (3 1) free Conjecture of Stanley and Stembridge [14]. 1 Introduction Let G be a finite graph with verticies V = V (G) fv 1 ; v 2 ; v d g and edge set E = E(G) We permit our graphs to have loops and multiple edges. Let XG (n) be the chromatic polynomial of G, i.e. the number of proper colorings : V f1; 2; ng. Proper means ....

....[8] In Section 3 we define YG and derive some of its basic properties, including the Deletion Contraction Law. Connections with acyclic orientations are explored in 1 Section 4. The next three sections are devoted to making some progress on the (3 1) free Conjecture of Stanley and Stembridge [14]. Finally we end with some comments and open questions. 2 Noncommutative symmetric functions Our noncommutative symmetric functions will be indexed by elements of the partition lattice. We let Pi d denote the lattice of set partitions of f1; 2; dg : d] ordered by refinement. We ....

[Article contains additional citation context not shown here]

R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory (A) 62 (1993), 261-- 279.

....graph and let XG be its chromatic symmetric function. Expand XG in terms of elementary symmetric functions e and call the coefficients a G : XG = X a G e : One of the outstanding conjectures about XG (and the motivation for this note) is the Stanley Stembridge Poset Chain Conjecture [6]: if G is a clawfree incomparability graph, then a G 0 for all . It is natural to attack this conjecture by looking for a combinatorial interpretation of a G . We can obtain such an interpretation as follows. Observe first that Gasharov [2] tells us that if G is a clawfree incomparability ....

.... to see that XG is a symmetric function, so it can be written as a (finite) linear combination of elementary symmetric functions e : XG = X d a G e : As we mentioned in the introduction, one of the main open problems in this area is the Poset Chain Conjecture of Stanley and Stembridge [6]. This states that if G is a clawfree incomparability graph, then G is e positive, i.e. a G 0 for all . Recall that an incomparability graph is a graph obtained from a finite poset by letting the vertex set of the graph be the vertex set of the poset and connecting two vertices with an edge ....

R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory (A) 62 (1993), 261--279.

....are matrices that arise in the theory of symmetric functions, there is a rich array of combinatorial conjectures related to immanants. While it is tempting to give a summary of these beautiful conjectures, we shall restrict ourselves to describing just one of them, due to Stanley and Stembridge [S S], since it is the only one directly relevant to our present purposes. The interested reader is referred to [Gre] and [S S] for the full story. To state the Stanley Stembridge conjecture (also known as the Poset Chain Conjecture) we must first define a certain invariant XP of a poset P . A ....

....to immanants. While it is tempting to give a summary of these beautiful conjectures, we shall restrict ourselves to describing just one of them, due to Stanley and Stembridge [S S] since it is the only one directly relevant to our present purposes. The interested reader is referred to [Gre] and [S S] for the full story. To state the Stanley Stembridge conjecture (also known as the Poset Chain Conjecture) we must first define a certain invariant XP of a poset P . A coloring of a poset P is a map that sends each element of P to a positive integer (or color ) and whose fibers are totally ....

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R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory (A) 62 (1993), 261--279.

....subgraph consisting of one vertex connected to three other vertices (and no further edges) Is XG s positive A special class of (3 1) free posets for which Problem 21 is still open is the semiorders or unit interval orders, de ned e.g. in [89, Exer. 6. 30] It is not hard to deduce from [90] that an armative answer to Problem 17 implies that Problem 21 has an armative answer for semiorders. Even the following very special case of Problem 21 is open (see [80, pp. 190 191] De ne F n = i 1 ; i n x i 1 x i 2 x i n ; where i 1 ; i n ranges over all sequences of ....

R. Stanley and J. R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combinatorial Theory (A) 62 (1993), 261-279.

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R. Stanley and J. R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted positions, J. Combinatorial Theory (A) 62 (1993) 261--279.

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R. Stanley and J. R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted positions, J. Combinatorial Theory (A) 62 (1993) 261--279.

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R. P. Stanley and J. R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), 261--279. 13

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R. Stanley and J. R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combinatorial Theory (A) 62 (1993), 261--279.

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R. Stanley and J. R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted positions, J. Combinatorial Theory (A) 62 (1993) 261--279.

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R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), 261--279.

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R. Stanley and J. Stembridge. On immanants of Jacobi-Trudi matrices and permutations with restricted position. J. Combin. Theory Ser. A, 62:261--279, 1993. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, U.S.A. E-mail address: lenart@math.mit.edu

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