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We introduce generalized permutation patterns, where we allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We show that essentially all Mahonian permutation statistics in the literature can be written as linear combinations of such patterns. Almost all known Mahonian permutation statistics can be written as(More)
We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of finite presentations of groups with one end, where it gives an exponential bound on the(More)
Motivated by the Coxeter complex associated to a Coxeter system (W, S), we introduce a simplicial regular cell complex ∆(G, S) with a G-action associated to any pair (G, S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of ∆(G, S), and in particular the representations of G on(More)
We provide a polynomial time algorithm for computing the universal Gröbner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the state polyhedron of any member of the Hilbert scheme Hilb d n of n-long d-variate ideals, enabled by introducing the(More)
Certain necessary conditions on the face numbers and Betti numbers of sim-plicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced(More)