No abstract

No abstract

Abstract. Consider the algebra Q x 1 , x 2 , . . . 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.

We introduce several analogs of the Robinson-Schensted algorithm for skew Young tableaux. These correspondences provide combinatorial proofs of various identities involving f&,, the number of standard skew tableaux of shape L/p, and the skew Schur functions So..,,. For example, we are able to show bijectively that and 4 S;&)QdY)=C ~p;p(x)~~,p(Y) n (1-%Y,)Y. P 1. I 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.

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the Mo bius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a generalization of Stanley's theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. We also give some applications of this second main theorem, including the Tamari lattices.1997 Academic Press BOUNDED BELOW SETSIn a fundamental paper [25], Whitney showed how broken circuits could be used to compute the coefficients of the chromatic polynomial of a graph. In another seminal paper [20], Rota refined and extended Whitney's theorem to give a characterization of the Mo bius function of a geometric lattice. Then one of us [21] generalized Rota's result to a larger class of lattices. In this paper we will present a theorem for an arbitrary finite lattice that includes all the others as special cases. To do so, we shall need to replace the notion of a broken circuit by a new one which we call a bounded below set. Next we present some applications to lattices whose Mo bius functions had previously been computed but in a less simple or less combinatorial way: shuffle posets [13], non-crossing set partition lattices [15,19], and integer partitions under dominance order [5,6,12].

Abstract. 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.

No abstract