Einar Steingrimsson

The Coloring Ideal and Coloring Complex of a Graph

Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G.The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisner ring of a simplicial complex C. The h-vector of C is a certain transformation of the tailT(n) = nd − χ (n) of the chromatic polynomial χ of G. The combinatorial structure of the complex C is described explicitly and it is shown that the Euler characteristic of C equals the number of acyclic orientations of G.

Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

We prove that the Stanley-Wilf limit of any layered permutation pattern of length ? is at most 4?2, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern.We also conjecture that, for any k?0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n+1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most eπ2/3?13.001954.

Generalized permutation patterns - a short survey

An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.

Catalan continued fractions and increasing subsequences in permutations

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let ek(π) be the number of increasing subsequences of length k+1 in the permutation π. We prove that any Catalan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly infinite) linear combination of the eks. Moreover, there is an invertible linear transformation that translates between linear combinations of eks and the corresponding continued fractions. Some applications are given, one of which relates fountains of coins to 132-avoiding permutations according to number of inversions. Another relates ballot numbers to such permutations according to number of right-to-left maxima.

Mixed Volumes and Slices of the Cube

We give a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a refinement of the Eulerian numbers.

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Mobius function of an interval [@s,@p] in the poset of permutations ordered by pattern containment in the case where @p is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,...,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Mobius function in the case where @s and @p are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. We also show that the Mobius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Mobius function of an interval [@s,@p] of separable permutations is bounded by the number of occurrences of @s as a pattern in @p. Another consequence is that for any separable permutation @p the Mobius function of (1,@p) is either 0, 1 or -1.

The volume of relaxed Boolean-quadric and cut polytopes

For n ⩾ 2, the boolean quadric polytope Pn is the convex hull in d:=(n+12) dimensions of the binary solutions xixj = yij, for all i < j in N ≔ 1,2,. …,n. The polytope is naturally modeled by a somewhat larger polytope; namely, Ln the solution set of uij ⩽ xij, yij ⩽ xj, xi + xj ⩽ 1 + yij, yij ⩾ 0, for all i, j in N. In a first step toward seeing how well Ln approximates Pn we establish that the d-dimensional volume of Ln is 22n−dn!/(2n)!. Using a well-known connection between Pn and the ‘cut polytope’ of a complete graph n + 1 vertices, we also establish the volume of a relaxation of this cut polytope.

On the Topology of the Permutation Pattern Poset

Abstract The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Mobius function of decomposable permutations given by Burstein et al. [9] .

Number of cycles in the graph of 312-avoiding permutations

Abstract The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation π = π 1 π 2 ⋯ π n + 1 there is a directed edge from the standardization of π 1 π 2 ⋯ π n to the standardization of π 2 π 3 ⋯ π n + 1 . We give a formula for the number of cycles of length d in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.

A chromatic partition polynomial

Abstract A polynomial in two variables is defined by C n ( x , t ) = Σ π ∈ Π n x ( G π , x ) t | π | , where Πn is the lattice of partitions of the set {1, 2, …, n}, Gπ is a certain interval graph defined in terms of the partition gp, χ(Gπ, x) is the chromatic polynomial of Gπ and |π| is the number of blocks in π. It is shown that C n ( x , t ) = Σ k = 0 n t k Σ i = 0 k ( n − k n − 1 ) S ( n , i ) ( x ) i , where S(n, i) is the Stirling number of the second kind and (x)i = x(x − 1) ··· (x − i + 1). As a special case, Cn(−1, −t) = An(t), where An(t) is the nth Eulerian polynomial. Moreover, A n ( t ) = Σ π ∈ Π n a π t | π | where aπ is the number of acyclic orientations of Gπ.