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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)
The constraint-based approach to analysis of biochemical systems has emerged as a useful tool for rational metabolic engineering. Flux balance analysis (FBA) is based on the constraint of mass conservation; energy balance analysis (EBA) is based on non-equilibrium thermodynamics. The power of these approaches lies in the fact that the constraints are based(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 Hilbn of n-long d-variate ideals, enabled by introducing the(More)
In this article we find the threshold for simple connectivity of the random 2dimensional simplicial complexes Y (n, p) introduced by Linial and Meshulam [10] to be roughly p = n−1/2. One motivation for this is continuing the thread of probabilistic topology initiated by Linial and Meshulam [10], and even earlier by Erdős and Rényi [3]. (Other recent work(More)
A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, ∆lex — an operation that transforms a monomial ideal of S = k[xi : i ∈ N] that is finitely generated in each degree into a squarefree strongly stable ideal — is defined and studied. It is proved that (in contrast(More)
Certain necessary conditions on the face numbers and Betti numbers of simplicial 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)