Enumerative Results Research Articles

Pattern statistics in faro words and permutations

We study the distribution and the popularity of some patterns in k-ary faro words, i.e. words over the alphabet {1,2,…,k} obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the x-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Finally, we consider some interesting subclasses of faro words that are permutations, involutions, derangements, or subexcedent words.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field$k$, generalizing the counts that over${\mathbf {C}}$there are$27$lines, and over${\mathbf {R}}$the number of hyperbolic lines minus the number of elliptic lines is$3$. In general, the lines are defined over a field extension$L$and have an associated arithmetic type$\alpha$in$L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group$\operatorname {GW}(k)$of$k$,\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]where$\operatorname {Tr}_{L/k}$denotes the trace$\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over${\mathbf {C}}$and${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in$\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Permutations, tensor products, and Cuntz algebra automorphisms

We study the reduced Weyl groups of the Cuntz algebras On from a combinatorial point of view. Their elements correspond bijectively to certain permutations of nr elements, which we call stable. We mostly focus on the case r=2 and general n. A notion of rank is introduced, which is subadditive in a suitable sense. Being of rank 1 corresponds to solving an equation which is reminiscent of the Yang-Baxter equation. Symmetries of stable permutations are also investigated, along with an immersion procedure that allows to obtain stable permutations of (n+1)2 objects starting from stable permutations of n2 objects. A complete description of stable transpositions and of stable 3-cycles of rank 1 is obtained, leading to closed formulas for their number. Other enumerative results are also presented which yield lower and upper bounds for the number of stable permutations.

Encoding Labelled p-Riordan Graphs by Words and Pattern-Avoiding Permutations

The notion of a p-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a p-Riordan word, and show how to encode p-Riordan graphs by p-Riordan words. For special important cases of Riordan graphs (the case $$p=2$$ ) and oriented Riordan graphs (the case $$p=3$$ ) we provide alternative encodings in terms of pattern-avoiding permutations and certain balanced words, respectively. As a bi-product of our studies, we provide an alternative proof of a known enumerative result on closed walks in the cube.

Pattern-avoiding permutation powers

Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation π strongly avoids a pattern τ if π and π2 both avoid τ. They conjectured that for every positive integer k, there is a permutation in Sk3 that strongly avoids 123⋯(k+1). We use the Robinson–Schensted–Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least kk3∕2+O(k3∕logk) and at most k2k3+O(k3∕logk). We enumerate 231-avoiding permutations of order 3, and we give two further enumerative results concerning strong pattern avoidance. We also consider permutations whose powers all avoid a pattern τ. Finally, we study subgroups of symmetric groups whose elements all avoid certain patterns. This leads to several new open problems connecting the group structures of symmetric groups with pattern avoidance.

An insertion algorithm on multiset partitions with applications to diagram algebras

We generalize the Robinson–Schensted–Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson [15].

A partial order on Motzkin paths

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a possibility by defining and studying an analogy of the Tamari lattice on Motzkin paths. While our generalization is not a lattice, each of its connected components is isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also extend the structural and enumerative results to Schröder paths. We conclude by a discussion on the relation between our work and that of Baril and Pallo (2014).

$k$-Arrangements, Statistics, and Patterns

The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blit...

On Foliations by Curves with Singularities of Positive Dimension

We present enumerative results for holomorphic foliations by curves on $\mathbb {P}^{n}$, n ≥ 3, with singularities of positive dimension. Some of the results presented improve previous ones due to Correa et al. (Annales de l’institut Fourier, 64(4):1781–1805, 2014) and Costa (Ann Fac Sci Toulouse, Math (6), 15(2):297–321, 2006). We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on $\mathbb {P}^{n}$ with a singularity of positive dimension. Moreover, we construct a family of holomorphic foliations by curves on $\mathbb {P}^{n}$ with a singularity of a positive dimension where its Milnor number is exhibited.

Counting Water Cells in Pattern Restricted Compositions

In this paper, we consider statistics on compositions of a positive integer represented geometrically as bargraphs that avoid certain classes of consecutive patterns. A unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within its subtended area is called a water cell since it is a place where a liquid would collect if poured along the top part of the bargraph from above. The total number of water cells in the bargraph representation of a k-ary word then gives what is referred to as the capacity of w. Here, we determine the distribution of the capacity statistic on certain pattern-restricted compositions, regarded as k-ary words. Several general classes of patterns are considered, including and where a is arbitrary. As a consequence of our results, we obtain all of the distinct distributions for the capacity statistic on avoidance classes of compositions corresponding to 3-letter patterns having at most two distinct letters. Finally, in the case of some further enumerative results are given when a=2, including algebraic and bijective proofs for the total capacity of all Carlitz partitions of a given size having a fixed number of blocks.

Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees

Given a gene tree topology and a species tree topology, a coalescent history represents a possible mapping of the list of gene tree coalescences to associated branches of a species tree on which those coalescences take place. Enumerative properties of coalescent histories have been of interest in the analysis of relationships between gene trees and species trees. The simplest enumerative result identifies a bijection between coalescent histories for a matching caterpillar gene tree and species tree with monotonic paths that do not cross the diagonal of a square lattice, establishing that the associated number of coalescent histories for n-taxon matching caterpillar trees (n⩾2) is the Catalan number Cn−1=1n(2n−2n−1). Here, we show that a similar bijection applies for non-matching caterpillars, connecting coalescent histories for a non-matching caterpillar gene tree and species tree to a class of roadblocked monotonic paths. The result provides a simplified algorithm for enumerating coalescent histories in the non-matching caterpillar case. It enables a rapid proof of a known result that given a caterpillar species tree, no non-matching caterpillar gene tree has a number of coalescent histories exceeding that of the matching gene tree. Additional results on coalescent histories can be obtained by a bijection between permissible roadblocked monotonic paths and Dyck paths. We study the number of coalescent histories for non-matching caterpillar gene trees that differ from the species tree by nearest-neighbor-interchange and subtree-prune-and-regraft moves, characterizing the non-matching caterpillar with the largest number of coalescent histories. We discuss the implications of the results for the study of the combinatorics of gene trees and species trees.

New scaling laws for self-avoiding walks: bridges and worms**Dedicated to the memory of Vladimir Rittenberg.

We show how the theory of the critical behaviour of d-dimensional polymer networks gives a scaling relation for self-avoiding bridges that relates the critical exponent for bridges to that of terminally-attached self-avoiding arches, and the correlation length exponent We find In the case of the special transition, we find We provide compelling numerical evidence for this result in both two- and three-dimensions. Another subset of SAWs, called worms, are defined as the subset of SAWs whose origin and end-point have the same x-coordinate. We give a scaling relation for the corresponding critical exponent which is This too is supported by enumerative results in the two-dimensional case.

On Partially Ordered Patterns of Length 4 and 5 in Permutations

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length $k$ is defined by a partially ordered set on $k$ elements, and classical patterns correspond to $k$-element chains. The notion of a POP provides a convenient language to deal with larger sets of permutation patterns.
 This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain 13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

Enumerating several aspects of non-decreasing Dyck paths

A Dyck path is non-decreasing if the y-coordinates of its valleys form a non-decreasing sequence. In this paper we give enumerative results and some statistics of several aspects of non-decreasing Dyck paths. We give the number of pyramids at a fixed level that the paths of a given length have, count the number of primitive paths, count how many of the non-primitive paths can be expressed as a product of primitive paths, and count the number of paths of a given height and a given length. We present and prove our results using combinatorial arguments, generating functions (using the symbolic method) and parameterize the results studied here using the Riordan arrays. We use known bijections to connect direct column-convex polyominoes, Elena trees, and non-decreasing Dyck paths.

Split Graphs: Combinatorial Species and Asymptotics

A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to Bína and Přibil (2015), Cheng, Collins, and Trenk (2016), and Collins and Trenk (2018). In both the labeled and unlabeled cases, we give asymptotic results on the number of split graphs, of unbalanced split graphs, and of bicolored graphs, including proving the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are balanced.

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from “partitioning” wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented.

An inversion statistic for reduced words

We study the graph on reduced words with edges given by the Coxeter relations for the symmetric group. We define a statistic on reduced words for a given permutation, analogous to Coxeter length for permutations, for which the graph becomes ranked with unique maximal element. We show this statistic extends naturally to balanced labellings, and use it to recover enumerative results of Edelman and Greene and of Reiner and Roichman.

Forcibly-biconnected Graphical Degree Sequences: Decision Algorithms and Enumerative Results

We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length $n$ and forcibly biconnected graphical partitions of given even integer $n$. Based on these enumerative results we make some conjectures such as: when $n$ is large, (1) the proportion of forcibly biconnected graphical degree sequences of length $n$ among all zero-free graphical degree sequences of length $n$ is asymptotically a constant $C$ ($0

A poset structure on the alternating group generated by 3-cycles

We investigate the poset structure on the alternating group that arises when the latter is generated by 3-cycles. We study intervals in this poset and give several enumerative results, as well as a complete description of the orbits of the Hurwitz action on maximal chains. Our motivating example is the well-studied absolute order arising when the symmetric group is generated by transpositions, i.e. 2-cycles, and we compare our results to this case along the way. In particular, noncrossing partitions arise naturally in both settings.

Simultaneous core multipartitions

We initiate the study of simultaneous core multipartitions, generalising simultaneous core partitions, which have been studied extensively in the recent literature. Given a multipartition datum(s∣c), which consists of a non-negative integer s and an l-tuple c of integers, we introduce the notion of an (s∣c)-core multipartition. Given an arbitrary set of multipartition data, we give necessary and sufficient conditions for the corresponding set of simultaneous core multipartitions to be finite. We then study the special case of simultaneous core bipartitions, giving exact enumerative results in some special subcases.