Bruce E. Sagan

The Symmetric Group

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The twisted N-cube with application to multiprocessing

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The Wiener Polynomial of a Graph

It is shown that by exchanging any two independent edges in any shortest cycle of the n-cube (n>or=3), its diameter decreases by one unit. This leads to the definition of a new class of n-regular graphs, denoted TQ/sub n/, with 2/sup n/ vertices and diameter n-1, which has the (n-1)-cube as subgraph. Other properties of TQ/sub n/ such as connectivity and the lengths of the disjoints paths are also investigated. Moreover, it is shown that the complete binary tree on 2/sup n/-1 vertices, which is not a subgraph of the n-cube, is a subgraph of TQ/sub n/. How these results can be used to enhance hypercube multiprocessors is discussed. >

Shifted tableaux, Schur Q -functions, and a conjecture of R. Stanley

rn The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincar6 polynomial of a finite Coxeter group. 0 1996 John Wiley & Sons, Inc.

A Littlewood-Richardson rule for factorial Schur functions

Abstract We present an analog of the Robinson-Schensted correspondence that applies to shifted Young tableaux and is considerably simpler than the one proposed in [ B. E. Sagan, J. Combin. Theory Ser. A 27 (1979) , 10–18]. In addition, this algorithm enjoys many of the important properties of the original Robinson-Schensted map including an interpretation of row lengths in terms of k-increasing sequences, a jeu de taquin, and a generalization to tableaux with repeated entries analogous to Knuths construction ( Pacific J. Math. 34 (1970) , 709–727). The fact that the Knuth relations hold for our algorithm yields a simple proof of a conjecture of Stanley.

Robinson-Schensted algorithms for Skew tableaux

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

A note on independent sets in trees

Abstract We introduce several analogs of the Robinson-Schensted algorithm for skew Young tableaux. These correspondences provide combinatorial proofs of various identities involving f λ μ the number of standard skew tableaux of shape λ μ , and the skew Schur functions s λ μ (x). For example, we are able to show bijectively that and It is then shown that these new algorithms enjoy some of the same properties as the original. In particular, it is still true that replacing a permutation by its inverse exchanges the two output tableaux. This fact permits us to derive a number of other identities as well.

Symmetric functions in noncommuting variables

We give a simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by \[ m( T ) = \begin{cases} 2^{k - 1} + 1& {\text{if }} n = 2k, \\ 2^k & {\text{if }} n = 2k + 1, \end{cases}\] a result first proved by Wilf [SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 125–130]. We also characterize those trees achieving this maximum value. Finally we investigate some related problems.

Representations of the Symmetric Group

Consider the algebra Q >of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x 1 , x 2 ,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

Inductive and injective proofs of log concavity results

In this chapter we construct all the irreducible representations of the symmetric group. We know that the number of such representations is equal to the number of conjugacy classes (Proposition 1.10.1), which in the case of S n is the number of partitions of n. It may not be obvious how to associate an irreducible with each partition λ = (λ1, λ2,...., λl), but it is easy to find a corresponding subgroup S λ that is an isomorphic copy of S λl x Sλ2 x · · · x S λl, inside S n . We can now produce the right number of representations by inducing the trivial representation on each Sλ up to S n .