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 composition poset", where they studied a poset where there are additional cover relations which allows the insertion of a part of size one anywhere in the composition. Finally, following a suggestion by Richard Stanley we study yet a third which is an extension of the previous two posets. This poset is related to quasi-symmetric functions.
For these posets, we study generating functions for saturated chains of fixed width k. We also construct ``labeled'' non-commutative generating functions and their associated languages.
A partition of a finite poset into chains places a natural upper bound on the size of a union of k antichains. A chain partition is k-saturated if this bound is achieved. Greene and Kleitman proved that, for each k, every finite poset has a simultaneously k- and k+1-saturated chain partition. West showed that the Greene-Kleitman Theorem is best-possible in a strong sense by exhibiting, for each c \ge 4, a poset with longest chain of cardinality c and no k- and l-saturated chain partition for any distinct, nonconsecutive k,l < c. We call such posets polyunsaturated. We give necessary and suffic…
Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation numbe…
We classify finite posets with a particular sorting property, generalizing a result for rectangular arrays. Each poset is covered by two sets of disjoint saturated chains such that, for any original labeling, after sorting the labels along both sets of chains, the labels of the chains in the first set remain sorted. We also characterize posets with more restrictive sorting properties.
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.
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…
Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a $P$-saturated family is denoted by $La(n,P)$, which has been studied for a number of choices of $P$. The minimum size of a $P$-saturated family, $sat(n,P)$, was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs. We introduce and study the concept of saturation for induced subposets. As opposed to in…
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…
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…
A poset has the non-messing-up property if it has two covering sets of disjoint saturated chains so that for any labeling of the poset, sorting the labels along one set of chains and then sorting the labels along the other set yields a linear extension of the poset. The linear extension yielded by thus twice sorting a labeled non-messing-up poset may be independent of which sort was performed first. Here we characterize such sort-invariant labelings for convex subposets of a cylinder. They are completely determined by avoidance of a particular subpattern: a diamond of four elements whose small…
A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $P$ is any symmetric chain order, we prove that $P^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $P^n$ by cyclic permutation of the factors.
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.
We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.
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…
Gessel's fundamental and Stembridge's peak functions are the generating functions for (enriched) $P$-partitions on labelled chains. They are also the bases of two significant subalgebras of formal power series, respectively the ring of quasisymmetric functions (QSym) and the algebra of peaks. Hsiao introduced the monomial peak functions, a basis of the algebra of peaks indexed by odd integer compositions whose relation to peak functions mimics the one between the monomial and fundamental bases of QSym. We show that the extension of monomial peaks to any composition is a new basis of QSym and g…
Given a finite graded poset with labeled Hasse diagram, we construct a quasi- symmetric generating function for (saturated) chains whose labels have fixed descents. This is a common generalization of a generating function for the flag f-vector defined by Ehrenborg and of a symmetric function associated to certain edge-labeled posets which arose in the theory of Schubert polynomials. We show this construction gives a Hopf morphism from an incidence algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions.
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.
This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classi…
We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all relations between ribbon Schur Q-functions; show they supply a Z-basis for skew Schur Q-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur Q-functions is 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 c not equal to c'. If we keep in place the latter axiom but allow for more than one morphism between objects, we can have a sort of generalized poset in which there are multiplicities attached to the 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, which incorporates earlier notion…
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…
Let P be a poset and let P* be the set of all finite length words over P. Generalized subword order is the partial order on P* obtained by letting u \leq w if and only if there is a subword u' of w having the same length as u such that each element of u is less than or equal to the corresponding element of u' in the partial order on P. Classical subword order arises when P is an antichain, while letting P be a chain gives an order on compositions. For any finite poset P, we give a simple formula for the Mobius function of P* in terms of the Mobius function of P. This permits us to rederive in …
