Subword Complexity Research Articles

Improved Prediction of Regulatory Element Using Hybrid Abelian Complexity Features with DNA Sequences.

Deciphering the code of cis-regulatory element (CRE) is one of the core issues of current biology. As an important category of CRE, enhancers play crucial roles in gene transcriptional regulations in a distant manner. Further, the disruption of an enhancer can cause abnormal transcription and, thus, trigger human diseases, which means that its accurate identification is currently of broad interest. Here, we introduce an innovative concept, i.e., abelian complexity function (ACF), which is a more complex extension of the classic subword complexity function, for a new coding of DNA sequences. After feature selection by an upper bound estimation and integration with DNA composition features, we developed an enhancer prediction model with hybrid abelian complexity features (HACF). Compared with existing methods, HACF shows consistently superior performance on three sources of enhancer datasets. We tested the generalization ability of HACF by scanning human chromosome 22 to validate previously reported super-enhancers. Meanwhile, we identified novel candidate enhancers which have supports from enhancer-related ENCODE ChIP-seq signals. In summary, HACF improves current enhancer prediction and may be beneficial for further prioritization of functional noncoding variants.

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that–the Thue–Morse word has the minimum possible subword complexity over all overlap-free binary words and all (73)-power-free binary words, but not over all (73)+-power-free binary words;–the twisted Thue–Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all (73)-power-free binary words;–if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word;–the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all symmetric square-free ternary words.

Automatic complexity of shift register sequences

Let x be an m-sequence, a maximal length sequence produced by a linear feedback shift register. We show that x has maximal subword complexity function in the sense of Allouche and Shallit. We show that this implies that the nondeterministic automatic complexity AN(x) is close to maximal: n∕2−AN(x)=O(log2n), where n is the length of x. In contrast, Hyde has shown AN(y)≤n∕2+1 for all sequences y of length n.

Maximal Repetition and Zero Entropy Rate

Maximal repetition of a string is the maximal length of a repeated substring. This paper investigates maximal repetition of strings drawn from stochastic processes. Strengthening previous results, two new bounds for the almost sure growth rate of maximal repetition are identified: an upper bound in terms of conditional R\'enyi entropy of order $\gamma>1$ given a sufficiently long past and a lower bound in terms of unconditional Shannon entropy ($\gamma=1$). Both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time. We also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity. In particular, it is shown that a power-law logarithmic growth of maximal repetition with respect to the string length, recently observed for texts in natural language, may hold only if the conditional R\'enyi entropy rate given a sufficiently long past equals zero. According to this observation, natural language cannot be faithfully modeled by a typical hidden Markov process, which is a class of basic language models used in computational linguistics.

Subword complexes via triangulations of root polytopes

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $\beta$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polytopes in terms of $\beta$-Grothendieck polynomials.

Towards a uniform subword complex description of acyclic finite type cluster algebras

It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. We continue this study by uniformly describing the c- and g-vectors, and by providing a conjectured description of the Newton polytopes of the F-polynomials. We moreover show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F-polynomials or of the cluster variables, respectively. We prove this conjectured description to hold in type A and in all types of rank at most 8 including all exceptional types, leaving types B, C, and D conjectural.

ν-Tamari lattices via subword complexes

We show that the ν-Tamari lattices of Préville-Ratelle and Viennot can obtained as the duals of certain subword complexes. This generalizes a known result for the classical Tamari lattice, provides a simple description of the lattice property using certain bracket vectors of ν-trees, and gives (conjectural) insight on the geometry of more general objects in terms of polytopal subdivisions of multiassociahedra.

Growth of monomial algebras, simple rings and free subalgebras

We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised in [9, Question 5.1].Our construction goes through a construction of finitely generated just-infinite, primitive monomial algebras with prescribed growth type, from which we construct uniformly recurrent infinite words with subword complexity having the same growth type.We also discuss the connection between entropy of algebras and their homomorphic images, as well as the degrees of their generators of free subalgebras.

Fan Realizations for Some 2-Associahedra

ABSTRACTA k-associahedron is a simplicial complex whose facets, called k-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no k + 1 diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes, and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This article reports on conjectural realizations for all 2-associahedra, obtained by heuristic methods arising from natural geometric intuition on subword complexes. Experiments certify that we obtain fan realizations of 2-associahedra of an n-gon for n ∈ {10, 11, 12, 13}, further ones being out of our computational reach.

Subword complexity and non-automaticity of certain completely multiplicative functions

In this article, we prove that for a completely multiplicative function f from N⁎ to a field K such that the set{p|f(p)≠1Kand p is prime} is finite, the asymptotic subword complexity of f is Θ(nt), where t is the number of primes p that f(p)≠0K,1K. This proves in particular that sequences like ((−1)v2(n)+v3(n))n are not k-automatic for k≥2.

A Hopf algebra of subword complexes

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.

Computing abelian complexity of binary uniform morphic words

Although a lot of research has been done on the factor complexity (also called subword complexity) of morphic words obtained as fixed points of iterated morphisms, there has been little development in exploring algorithms that can efficiently compute their abelian complexity. The factor complexity counts the number of factors of a given length n, while the abelian complexity counts that number up to letter permutation. We propose and analyze a simple O(n) algorithm for quickly computing the exact abelian complexities for all indices from 1 up to n, when considering binary uniform morphisms. Using our algorithm we also analyze the structure in the abelian complexity for that class of morphisms. Our main result implies, in particular, that the infinite word over the alphabet {−1,0,1} constructed from the consecutive forward differences of the abelian complexity of a fixed point of a binary uniform morphism is in fact an automatic sequence with the same morphic length. Since the proof produces morphisms that typically contain many redundant letters, we present an efficient algorithm to eliminate them in order to simplify the morphisms and to see the patterns produced more clearly.

Toric matrix Schubert varieties and their polytopes

Given a matrix Schubert variety $\overline{X_\pi}$, it can be written as $\overline{X_\pi}=Y_\pi\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_{\pi}$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $\Phi(\mathbb{P}(Y_\pi))$ of its projectivization. We construct regular triangulations of $\Phi(\mathbb{P}(Y_\pi))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

Brick Manifolds and Toric Varieties of Brick Polytopes

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

Enumeration and Automatic Sequences

Abstract A sequence (an)n≥0 is k-automatic if there is a finite automaton that, on input n expressed in base k, reaches a state with output an. In this paper I will survey some recent advances concerning enumeration of various aspects of these sequences, such as the recurrence function, and the subword complexity (which counts the number of distinct blocks of length n).

Fan Realizations of Type $$A$$ A Subword Complexes and Multi-associahedra of Rank 3

We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$.

Subword Complexes and Nil-Hecke Moves

For a finite Coxeter group W, a subword complex is a simplicial complex associated with a pair (Q, ρ), where Q is a word in the alphabet of simple reflections, ρ is a group element. We describe the transformations of such a complex induced by nil-moves and inverse operations on Q in the nil-Hecke monoid corresponding to W. If the complex is polytopal, we also describe such transformations for the dual polytope. For W simply-laced, these descriptions and results of [5] provide an algorithm for the construction of the subword complex corresponding to (Q, ρ) from the one corresponding to (δ(Q), ρ), for any sequence of elementary moves reducing the word Q to its Demazure product δ(Q). The former complex is spherical or empty if and only if the latter one is empty. The article is published in the author’s wording.

Brick polytopes of spherical subword complexes and generalized associahedra

We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, a Minkowski sum decomposition into Coxeter matroid polytopes, and a combinatorial description of the exchange matrix of any cluster in a finite type cluster algebra.

Fan realizations of type $A$ subword complexes and multi-associahedra of rank 3

We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$. Nous construisons des éventails simpliciaux complets ayant la combinatoire des complexes de sous-mots de type $A_n$ pour $n\leq 3$. Par conséquent, nous obtenons des constructions d’éventails des multi-associaèdres $\Delta_{2k+4,k}$, dont les facettes correspondent aux $k$-triangulations d’un $(2k+4)$-gone. Cette construction confirme l’existence d’éventails ayant la combinatoire du multi-associaèdres pour une famille dont la polytopalité n’est pas confirmée. Les techniques utilisées fonctionnent pour tous les groupes de Coxeter et nous espérons qu’elles seront utiles afin de construire des éventails réalisant les complexes de sous-mots en général. En particulier, nous présentons des éventails pour deux complexes de sous-mots de type $A_4$ dont l’existence était inconnue: les multi-associaèdres $\Delta_{9,2}$ et $\Delta_{11,3}$.

Subword complexes and edge subdivisions

For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair (Q, π), where Q is a word in the alphabet of simple reflections and π is a group element. We discuss the transformations of such a complex that are induced by braid moves of the word Q. We show that under certain conditions, such a transformation is a composition of edge subdivisions and inverse edge subdivisions. In this case, we describe how the H- and γ-polynomials change under the transformation. This case includes all braid moves for groups with simply laced Coxeter diagrams.