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.
The $Γ$-colored $d$-complete and $Γ$-colored minuscule posets unify and generalize multiple classes of colored posets introduced by R.A. Proctor, J.R. Stembridge, and R.M. Green. In previous work, we showed that $Γ$-colored minuscule posets are necessary and sufficient to build from colored posets certain representations of Kac--Moody algebras that generalize minuscule representations of semisimple Lie algebras. In this paper we classify $Γ$-colored minuscule posets, which also classifies the corresponding representations. We show that $Γ$-colored minuscule posets are precisely disjoint unions…
The $Γ$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $Γ$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $Γ$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $Γ$-colored $d$-complete posets are precisely order filters of the connected full hea…
The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach we introduce colored analogs of $P$-partitions and enriched $P$-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile's theory of combinatorial Hopf algebras and its colored analog.
We introduce the following class of partizan games, called pomax games. Given a partially ordered set whose elements are colored black or white, the players Black and White take turns removing any maximal element of their own color. If there is no such element, the player loses. We prove that pomax games are always integer-valued and for colored tree posets and chess-colored Young diagram posets we give a simple formula for the value of the game. However, for pomax games on general posets of height 3 we show that the problem of deciding the winner is PSPACE-complete and for posets of height 2 …
R.M. Green described structural properties that ``doubly infinite'' colored posets should possess so that they can be used to construct representations of most affine Kac--Moody algebras. These representations are analogs of the minuscule representations of the semisimple Lie algebras, and his posets (``full heaps'') are analogs of the finite minuscule posets. Here only simply laced Kac--Moody algebras are considered. Working with their derived subalgebras, we provide a converse to Green's theorem. Smaller collections of colored structural properties are also shown to be necessary and sufficie…
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…
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 bec…
A superdiagonal composition is one in which the $i$-th part or summand is of size greater than or equal to $i$. In this paper, we study the number of palindromic superdiagonal compositions and colored superdiagonal compositions. In particular, we give generating functions and explicit combinatorial formulas involving binomial coefficients and Stirling numbers of the first kind.
We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and quaternary words, spanning trees in the ladder graph, unimodal sequences covering an initial interval, and ordered-consecutive partition sequences. Our approach relies on the basic idea of representing the colored compositions as tilings of an $n$-board whose tiles are connected, and sometimes decorated, according to a suitable combinatorial interpretation of the giv…
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…
The incidence category $\C_{\F}$ of a family $\F$ of colored posets closed under disjoint unions and the operation of taking convex sub-posets was introduced by the author in \cite{Sz}, where the Ringel-Hall algebra $\H_{\F}$ of $\C_{\F}$ was also defined. We show that if the Hasse diagrams underlying $\F$ are rooted trees, then the subspace $\n_{\F}$ of primitive elements of $\H_{\F}$ carries a pre-Lie structure, defined over $\mathbb{Z}$, and with positive structure constants. We give several examples of $\n_{\F}$, including the nilpotent subalgebras of $\mathfrak{sl}_n$, $L \mathfrak{gl}_n$…
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.
This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number of colored compositions for any coloration. The interesting consequences arising from this relationship also give an immediate and simple criterion to determine whether a sequence of integers counts the number of some colored compositions. Applications to Catalan and Fibonacci numbers naturally emerge, allowing to clearly answer to some open questions. Moreo…
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.
For a given integer $d\ge 1$, we consider $\binom{n+d-1}{d}$-color compositions of a positive integer $ν$ for which each part of size $n$ admits $\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such compositions, generalizing existing results for $n$-color compositions (case $d=1$) and $\binom{n+1}{2}$-color compositions (case $d=2$). In addition, we give bijections from the set of $\binom{n+d-1}{d}$-color compositions of $ν$ to the set of compositions of $(d+1)ν- 1$ having only parts of size $1$ and $d+1$, the set of compositions of $(d+1)ν$ having only parts of siz…
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…
