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 Möbius 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.
We study filters in the partition lattice formed by restricting to partitions by type. The Möbius function is determined in terms of the easier-to-compute descent set statistics on permutations and the Möbius function of filters in the lattice of integer compositions. When the underlying integer partition is a knapsack partition, the Möbius function on integer compositions is determined by a topological argument. In this proof the permutahedron makes a cameo appearance.
A cascade of dihedral symmetries is hidden in Young's lattice of integer partitions. In fact, for each integer N>2 the Hasse graph of the subposet consisting of the partitions with maximal hook length strictly less than N has the dihedral group of order 2N as its symmetry group. Here a new interpretation of those Hasse graphs is presented, namely as the 1-skeleta of the injective hulls of certain finite metric spaces.
We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the Möbius number of the lattice of cycles of the matroid up to the sign. The purpose has been to find a model to establish an analogous result for the case of $q$-matroids and we find a relation between the Euler characteristic of the simplicial chain complex associated to a $q$-matroid complex and the lattice of $q$-cycles of the $q$-matroid. We use this formula to find the complete homology over $\mathbb{Z}$ of this shellable simplicial complex. We give a characterization of …
We compute the Möbius function for the subgroup lattice of the simple Suzuki group $Sz(q)$; this is used to enumerate normal subgroups of certain Hecke groups with quotients isomorphic to $Sz(q)$.
Closed-form expressions are obtained for the generating function of close-packed dimers on a $2M \times 2N$ simple quartic lattice embedded on a Möbius strip and a Klein bottle. Finite-size corrections are also analyzed and compared with those under cylindrical and free boundary conditions. Particularly, it is found that, for large lattices of the same size and with a square symmetry, the number of dimer configurations on a Möbius strip is 70.2% of that on a cylinder. We also establish two identities relating dimer generating functions for Möbius strips and cylinders.
We give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Mobius function, recovering a classical result due to Verma.
The energy of a simple graph $G$ arising in chemical physics, denoted by $\mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if $\{G_n\}$ is a sequence of finite simple graphs with bounded average degree and $\{G_n'\}$ a sequence …
A certain inequality is shown to hold for the values of the Mobius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen-Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley's nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen-Macaulay meet-semilattice and Adin's nonnegativity conjecture for the cubical h-vector of a Cohen-Macaulay cubical complex.
There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the Möbius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbi…
A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Mobius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.
The invertibility of LCM matrices and their Hadamard powers have been studied a lot over the years by many authors. Bourque and Ligh conjectured in 1992 that the LCM matrix $[S]=[[x_i, x_j]]$ on any GCD closed set $S=\{x_1, x_2, \ldots, x_n\}$ is invertible, but in 1997 this was proven false. However, currently there are many open conjectures concerning LCM matrices and their Hadamard powers presented by Hong. In this paper we utilize lattice-theoretic structures and the Möbius function to explain the singularity of classical LCM matrices and their Hadamard powers. At the same time we end up d…
In important work on the parity of the partition function, Ono related values of the partition function to coefficients of a certain mock theta function modulo 2. In this paper, we use Möbius inversion to give analogous results which relate several combinatorial functions via identities rather than congruences.
In these notes we study several categorical generalizations of the Möbius function and discuss the relations between the various approaches. We emphasize the topological and geometric meaning of these constructions.
A permutation τcontains another permutation σas a pattern if τhas a subsequence whose elements are in the same order with respect to size as the elements in σ. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when σoccurs precisely once in τ, and σand τsatisfy certain further conditions, in which case the Mobius function is shown to be either -1, 0 or 1. We conjecture that for intervals [σ,τ] consisting of permutations avoid…
This paper concerns the lattice $L_n$ of subsets of $\{1,\ldots,n\}$ that are arithmetic progressions, under the inclusion order. For $n\geq 4$, this poset is not graded and thus not semimodular. We give three independent proofs of the fact that for $n\geq 2$, $μ_n(L_n) = μ(n-1)$, where $μ_n$ is the Möbius function of $L_n$ and $μ$ is the classical (number-theoretic) Möbius function. We also show that $L_n$ is comodernistic, which implies that $L_n$ is EL-labelable. Comodernism is then used to prove that the order complex $Δ_n$ of the lattice is either contractible or homotopy equivalent to a …
