Möbius Functions of Lattices

Abstract

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].

Applications of quotient posets

“…By assumption (2), all the atomic transversals have the same support size which is the rank of x. It follows that each atomic transversal for x has Möbius value (−1)…”

“…First, we need to show the quotient is a homogeneous quotient. Conditions (2) and (3) in the definition of a transversal function (Definition 3.2) imply condition (1) of a homogeneous quotient (Definition 2.2). To show condition (2) holds, suppose that T x ≤ T y .…”

“…Having shown what Theorem 4.4 can say about the Tamari lattices, we now turn our attention to seeing how it implies a theorem of [2]. To explain both the theorem as well as how to prove it, we begin by defining the notion of a partition of an atom set being induced by a multichain.…”

“…In order to calculate the characteristic polynomial for T 3 we need a function, ρ. We will use generalized rank which was introduced in [2]. To define generalized rank, let us set up some notation.…”

Circular planar electrical networks: Posets and positivity

“…By [BS,Proposition 2.3], we have µ( 0, 1) = (−1) n−1 C n−1 in N C n , and furthermore, for any π ∈ N C n , the interval ( 0, π) is isomorphic to a product of the partition lattices N C k , where k ranges over the block sizes of π. Given π ∈ N C n , π may be represented as a set of dividing lines in a disk D with boundary vertices V 1 , V 2 , .…”