Bergeron, Bousquet-Melou and Dulucq enumerated paths in the Hasse diagram of the following poset: the underlying set is that of all compositions, and a composition μcovers another composition λif μcan be obtained from λby adding 1 to one of the parts of λ, or by inserting a part of size 1 into λ.
We employ the methods they developed in order to study the same problem for the following poset: the underlying set is the same, but μcovers λif μcan be obtained from λby adding 1 to one of the parts of λ, or by inserting a part of size 1 at the left or at the right of λ. This poset is of interest because of its relation to non-commutative term orders.
We generalize Björner and Stanley's poset of compositions to $m$-colored compositions. Their work draws many analogies between their (1-colored) composition poset and Young's lattice of partitions, including links to (quasi-)symmetric functions and representation theory. Here we show that many of these analogies hold for any number of colors. While many of the proofs for Björner and Stanley's poset were simplified by showing isomorphism with the subword order, we remark that with 2 or more colors, our posets are not isomorphic to a subword order.
We study three different poset structures on the set of all compositions. In the first case, the covering relation consists of inserting a part of size one to the left or to the right, or increasing the size of some part by one. The resulting poset was studied by the author in "A poset classifying non-commutative term orders", and then in "Standard paths in another composition poset" where some results about generating functions for standard paths in this poset was established. The latter article was inspired by the work of Bergeron, Bousquet-M{é}lou and Dulucq on "Standard paths in the compos…
We study a poset of compositions restricted by part size under a partial ordering introduced by Björner and Stanley. We show that our composition poset $C_{d+1}$ is isomorphic to the poset of words $A_d^*$. This allows us to use techniques developed by Björner to study the Möbius function of $C_{d+1}$. We use counting arguments and shellability as avenues for proving that the Möbius function is $μ(u,w)=(-1)^{|u|+|w|}{w\choose u}_{dn}$, where ${w\choose u}_{dn}$ is the number of $d$-normal embeddings of $u$ in $w$. We then prove that the formal power series whose coefficients are given by the z…
In 1982, Jónsson and McKenzie posed the following problem: "Find counter examples (or prove that none exist) to the refinement of $A^C\cong B^D$ [$A$, $B$, $C$, and $D$ non-empty posets] under" the condition "$C$, $D$, and $A^C$ are finite and connected." That is, in this situation, are there posets $E$, $X$, $Y$, and $Z$ such that $A\cong E^X$, $B\cong E^Y$, $C\cong Y\times Z$, and $D\cong X\times Z$? In this note, this problem is solved.
For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π^{\bullet}_{\vec{c}}$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition $\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^{B}$ where $B$ is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homol…
We introduce iposets---posets with interfaces---equipped with a novel gluing composition along interfaces and the standard parallel composition. We study their basic algebraic properties as well as the hierarchy of gluing-parallel posets generated from singletons by finitary applications of the two compositions. We show that not only series-parallel posets, but also interval orders, which seem more interesting for modelling concurrent and distributed systems, can be generated, but not all posets. Generating posets is also important for constructing free algebras for concurrent semirings and Kl…
We determine the Möbius function of the poset of compositions of an integer. In fact we give two proofs of this formula, one using an involution and one involving discrete Morse theory. The composition poset turns out to be intimately connected with subword order, whose Möbius function was determined by Björner. We show that using a generalization of subword order, we can obtain both Björner's results and our own as special cases.
A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $α=(α_1,\cdots,α_m)$, with $0\le α_i \le n$ for each $i=1,\cdots,m$. Define order $<$ as follow, $\forall α,β\in N(m,n)$, $β< α$ if and only if $β_i \le α_i(i=1,\cdots,m)$ and $\sum\limits_{i=1}^{m}β_i <\sum\limits_{i=1}^{m}α_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.
We consider the zeta and Möbius functions of a partial order on integer compositions first studied by Bergeron, Bousquet-Mélou, and Dulucq. The Möbius function of this poset was determined by Sagan and Vatter. We prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function and the Möbius function. The proofs are either directly from the definitions or by constructing finite-state automata. We also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones. We reprove Sagan…
By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered graphs when paired with the poset of compositions arising from the Pieri rules for quasisymmetric Schur functions and its deformation.
One of the central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM inequality into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.
Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive lattice L(alpha) consisting of all lower order ideals of F(alpha). These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that L(alpha) is rank unimodal. We show that if one of the…
A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c,c') and Hom(c',c) is nonempty for distinct objects c,c'. If we keep in place the latter axiom but allow for more than one morphism between objects, we have a sort of generalized poset in which there are multiplicities attached to covering relations, and possibly nontrivial automorphism groups. We call such a category an "updown category". In this paper we give a precise definition of such categories and develop a theory for them. We also give a detailed account of te…
A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erdős characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \subset \dotsc \subset A_k$ of an arbitrary length $k$. The extremal families contain all subsets whose cardinalities belong to an interval of length $k-1$ centred at $n/2$. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to app…
In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,n+1)-core partition with distinct parts. In particular, we conjectured that the distribution is asymptotically normal. However, our analysis hinged on a characterization of (n,n+1)-core partitions given by Straub, which is not readily generalized to other families of simultaneous core partitions. In another paper (arXiv:1611.05775) with Doron Zeilberger, we made use of the characterization in terms of posets to analyze (2n+1,2n+3)-core partit…
We give yet another proof for Faà di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer compositions, for which a Faà di Bruno-like formula is quite naturally established.
In this article, we take a closer look at the reliability of large minimal networks constructed by repeated compositions of the simplest possible networks. For a given number of devices $n=2^m$ we define the set of all the possible compositions of series and parallel networks of two devices. We then define several partial orders over this set and study their properties. As far as we know the ranking problem has not been addressed before in this context, and this article establishes the first results in this direction. The usual approach when dealing with reliability of two-terminal networks is…
