arXiv subjects

Einar Steingrímsson

Publications and source records attributed to Einar Steingrímsson.

7 recordsLinked to original sources

arXiv2026

Permutations, Pattern Covers, Posets and Positivity: Part I: Patterns and Posets

We study permutations covered by overlapping occurrences of a consecutive pattern. Special cases of this have been studied for consecutive patterns and via the cluster method. We study enumeration sequences obtained by repeating a fixed overlap. We elicit many structural properties of such pattern covers, mainly in terms of posets whose linear extensions are in bijection with the permutations in question. The enumeration of these permutations for different patterns gives rise to a great variety of sequences, many classical, but most of them not found in the OEIS. We conjecture that, essentially, all these sequences are moments of probability measures on the real line. We develop an enumerative and order-theoretic framework for this. For overlap 1 we give a generating function whose coefficients enumerate permutations covered by any fixed pattern with any fixed number of occurrences. For overlap at most half the pattern length we derive a polynomial-time enumeration formula. We associate to each pattern, overlap, and number of occurrences a poset whose linear extensions are the corresponding permutations. The regularity of this poset construction makes possible a structural analysis that in many cases leads to enumerative formulas for large classes of patterns. We prove uniform restrictions on the local structure of the Hasse diagrams, describe their branches (pendant chains) in terms of the corresponding patterns, and characterize when the poset is a tree or chain or subdivisions of a grid. Some of these constructions yield explicit families of formulas and explain unexpected equinumerations. In a forthcoming paper we investigate the sequences arising here from a probabilistic perspective, viewing them both as point processes and as moment sequences. We also obtain integral and matrix representations for important overlap 1 and 2 classes of pattern-covering sequences.

math.CO↗

arXiv2026

Involution $h$ on Catalan structures

We define an involution $h$ on Catalan structures through an abstract framework, prove an equidistribution theorem for four canonical statistics and present a generating function carrying these. This framework encompasses all combinatorial structures with a decomposition mirroring the first-return decomposition of Dyck paths. The fixed points of~$h$ are counted by Catalan numbers. Canonical bijections transport the equidistribution to eight well known concrete families, identifying the canonical statistics with native ones on each. In addition to its primary structure, each Catalan structure has a derived \emph{secondary structure}, and $h$~interchanges primary and secondary structure. The involution factors as $h = \rev \circ \corev \circ \rev$, where $\rev$ and $\corev$ are two simpler involutions, and the composition $M = h \circ \rev$ coincides with Donaghey's automorphism on plane trees. This yields $M^{-1} = \rev \circ M \circ \rev$ and a period theorem: Iterating the secondary structure construction produces a sequence that repeats with period equal to the order of~$M$. It is an open problem to describe $h$ and the canonical statistics explicitly on most of the more than two hundred known families of Catalan structures.

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arXiv2023

Permutations with few inversions

A curious generating function $S_0(x)$ for permutations of $[n]$ with exactly $n$ inversions is presented. Moreover, $(xC(x))^iS_0(x)$ is shown to be the generating function for permutations of $[n]$ with exactly $n-i$ inversions, where $C(x)$ is the generating function for the Catalan numbers.

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arXiv2020

Permutations, moments, measures

Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, in the form of a continued fraction, for a fourteen-parameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups. Special cases include several classical and noncommutative probability laws, along with a substantial subset of the orthogonalizing measures in the q-Askey scheme, now given a new combinatorial interpretation in terms of elementary permutation statistics. This framework further captures a variety of interesting combinatorial sequences including, notably, the moment sequences associated to distributions of the numbers of occurrences of (classical and vincular) permutation patterns of length three. This connection between pattern avoidance and broader ideas in classical and noncommutative probability is among several intriguing new corollaries, which generalize and unify results previously appearing in the literature, while opening up new lines of inquiry. The fourteen combinatorial statistics further generalize to signed and colored permutations, and, as an infinite family of statistics, to the k-arrangements: permutations with k-colored fixed points, introduced here along with several related results and conjectures.

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arXiv2020

Sorting with pattern-avoiding stacks: the $132$-machine

This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari [CCF]. These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed in [CCF]. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.

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arXiv2012

An involution on β(1,0)-trees

In [Decompositions and statistics for β(1,0)-trees and nonseparable permutations, Advances Appl. Math. 42 (2009) 313--328] we introduced an involution, h, on β(1,0)-trees. We neglected, however, to prove that h indeed is an involution. In this note we provide the missing proof. We also refine an equidistribution result given in the same paper.

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arXiv2011

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 $\ell$ is at most $4\ell^2$, 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. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also conjecture that, for any $k\ge 0$, 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 $e^{π\sqrt{2/3}} \simeq 13.001954$.

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