Poset Of Partitions Research Articles

On the poset of partitions of an integer

On the poset of partitions of an integer

Pointed and multi-pointed partitions of type A and B

The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we show that some operads are Koszul over \(\mathbb{Z}\).

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen–Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.

Hodge structures on posets

Let P P be a poset with unique minimal and maximal elements 0 ^ \hat {0} and 1 ^ \hat {1} . For each r r , let C r ( P ) C_r(P) be the vector space spanned by r r -chains from 0 ^ \hat {0} to 1 ^ \hat {1} in P P . We define the notion of a Hodge structure on P P which consists of a local action of S r + 1 S_{r+1} on C r C_r , for each r r , such that the boundary map ∂ r : C r → C r − 1 \partial _r: C_r\to C_{r-1} intertwines the actions of S r + 1 S_{r+1} and S r S_r according to a certain condition. We show that if P P has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of H r ( P ) H_r(P) into r r Hodge pieces. We consider the case where P P is B n , k \mathcal {B}_{n,k} , the poset of subsets of { 1 , 2 , … , n } \{1,2,\dots , n\} with cardinality divisible by k k ( k (k is fixed, and n n is a multiple of k ) k) . We prove a remarkable formula which relates the characters B n , k \mathcal {B}_{n,k} of S n S_n acting on the Hodge pieces of the homologies of the B n , k \mathcal {B}_{n,k} to the characters of S n S_n acting on the homologies of the posets of partitions with every block size divisible by k k .

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.

Partition complexes, duality and integral tree representations

We show that the poset of non-trivial partitions of 1,2,...,n has a fundamental homology class with coefficients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations of the symmetric groups S_n and S_{n+1} on the homology and cohomology of this partially-ordered set.

Noncrossing Partitions for the GroupDn

The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W. It is a self-dual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2,...,n} defined by Kreweras in 1972 when W is the symmetric group Sn, and to its type B analogue defined by the second author in 1997 when W is the hyperoctahedral group. We give a combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type Dn, thus answering a question of Bessis. Using this description, we compute a number of fundamental enumerative invariants of this lattice, such as the rank sizes, number of maximal chains, and Mobius function. We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B, and C. This leads to a (case-by-case) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to W-orbits.

Chain decomposition and the flag f-vector

Ehrenborg introduced a quasi-symmetric function encoding, denoted F P , for the flag f-vector of any finite, graded poset P with 0 ̂ and 1 ̂ . Stanley observed that F P is a symmetric function whenever P is locally rank-symmetric and asked for conditions under which F P is Schur-positive. We provide formulas for F P for three classes of locally rank-symmetric posets: graded monoid posets, generalized posets of shuffles and noncrossing partition lattices for classical reflection groups. Our flag f-vector expressions for generalized shuffle posets and noncrossing partition lattices exhibit Schur-positivity, while graded monoid posets do not always have Schur-positive flag f-vector. Each of our flag f-vector expressions results from a poset chain decomposition. For the noncrossing partition lattices and shuffle posets, these also yield symmetric chain decompositions (by restriction to 1-chains), shellability and supersolvability results and combinatorial formulae including characteristic polynomial and zeta polynomial. Our (more complicated) flag f-vector expression for graded monoid posets involves Gröbner bases and a weighted notion of Möbius function for the poset of partitions of a multiset and related multiset intersection posets.

Integer partitions and the Sperner property

The objectives of this paper are three-fold. First, we would like to call attention to a very attractive problem, the question of whether or not the poset of integer partitions ordered by refinement has the Sperner property. We provide all necessary definitions, and enough bibliography to interest a newcomer in the problem. Second, we prove four new theorems, two by exhaustive computation and two in the more traditional manner. Finally, we highlight the central role played by Larry Harper in the literature of this subject.

An Analogue of Covering Space Theory for Ranked Posets

Suppose $P$ is a partially ordered set that is locally finite, has a least element, and admits a rank function. We call $P$ a weighted-relation poset if all the covering relations of $P$ are assigned a positive integer weight. We develop a theory of covering maps for weighted-relation posets, and in particular show that any weighted-relation poset $P$ has a universal cover $\tilde P\to P$, unique up to isomorphism, so that 1. $\tilde P\to P$ factors through any other covering map $P'\to P$; 2. every principal order ideal of $\tilde P$ is a chain; and 3. the weight assigned to each covering relation of $\tilde P$ is 1. If $P$ is a poset of "natural" combinatorial objects, the elements of its universal cover $\tilde P$ often have a simple description as well. For example, if $P$ is the poset of partitions ordered by inclusion of their Young diagrams, then the universal cover $\tilde P$ is the poset of standard Young tableaux; if $P$ is the poset of rooted trees ordered by inclusion, then $\tilde P$ consists of permutations. We discuss several other examples, including the posets of necklaces, bracket arrangements, and compositions.

On the topology of two partition posets with forbidden block sizes

We study two subposets of the partition lattice obtained by restricting block sizes. The first consists of set partitions of {1,…, n} with block size at most k, for k≤ n−2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2 k+2≥ n and n=3 k+2. For 2 k+2> n, the posets in fact have the same S n−1 -homotopy type as the order complex of Π n−1 , and the S n -homology representation is the “tree representation” of Robinson and Whitehouse. We present similar results for the subposet of Π n in which a unique block size k≥3 is forbidden. For 2 k≥ n, the order complex has the homotopy type of a wedge of ( n−4)-spheres. The homology representation of S n can be simply described in terms of the Whitehouse lifting of the homology representation of Π n−1 .

An upper bound for the size of the largest antichain in the poset of partitions of an integer

Let Pi n be the poset of partitions of an integer n, ordered by refinement. Let b( Pi n ) be the largest size of a level and d( Pi n ) be the largest size of an antichain of Pi n . We prove that d(Pi n) b(Pi n) ⩽ e+ o(1) as n→∞. The denominator is determined asymptotically. In addition, we show that the incidence matrices in the lower half of Pi n have full rank, and we prove a tight upper bound for the ratio from above if Pi n is replaced by any graded poset P.

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.

Partitions with Restricted Block Sizes, Möbius Functions, and the k-of-Each Problem

Given a list of n real numbers, one wants to decide whether every number in the list occurs at least k times. It will be shown that $\Omega(n\log n)$ is a sharp lower bound for the depth of an algebraic decision or computation tree solving this problem for a fixed k. For linear decision trees, the coefficient can be taken to be arbitrarily close to 1 (using the ternary logarithm). This is done by using the Björner--Lovász--Yao method, which turns the problem into one of estimating the Möbius function for a certain partition lattice. The method will work also for the more general T-multiplicity problem when T is additive and cofinite. A formula for the exponential generating function for the Möbius function of a partition poset with restricted block sizes in general will also be given.

On q-analogues of partially ordered sets

We define a notation of an f-q-analogue of a poset P, were f is a function in the incidence algebra of P. In this setting, for given f and q, all f-q-analogues have the same Zeta and characteristic polynomials, and the same Möbius invarints for rank selections. These are q-analogues of the corresponding entities for th eposet P. We describe conditions when P is being shellable implies that its f-q-analogues are also shellable. In such situations, the analogues admit a shelling pulled back in a natural way from one of P, revealing a natural projection from the homology of analogues to that of P. As a by-product we obtain the non-negativity of the coefficients of the Betti polynomial for the analogues and their rank-selected subposets. We discuss the behavior of f-q-analogues with respect to several operations on the function f, the value q, and the poset P. Examples include posets of set partitions, posets of shuffles, semimodular and distributive lattices, and products of chains in particular. This work is an attempt to unify recent approaches to order analogues, and integrates cases studied by Butler; Björner and Stanley; and Bennett, Dempsey and Sagan.

Direct Sum Decompositions of Matroids and Exponential Structures

We associate to a simple matroid (resp. a geometric lattice) M and a number d dividing the rank of M a partially ordered set D d ( M ) whose upper intervals are (set-) partition lattices. Indeed, for some important cases they are exponential structures in the sense of Stanley [11]. Our construction includes the partition lattice, the poset of partitions whose size is divisible by a fixed number d , and the poset of direct sum decompositions of a finite vector space. If M is a modularly complemented matroid the posets D d ( M ) are CL-shellable. This generalizes results of Sagan and Wachs and settles the open problem of the shellability of the poset of direct sum decompositions. We analyse the shelling and derive some facts about the descending chains. We can apply these techniques to retrieve the results of Wachs about descending chains in the lattice of d -divisible partitions. We also derive a formula for the Möbius number of the poset of direct sum decompositions of a vector space.

Linear decision trees, subspace arrangements and Möbius functions

Topological methods are described for estimating the size and depth of decision trees where a linear test is performed at each node. The methods are applied, among others, to the questions of deciding by a linear decision tree whether given n n real numbers (1) some k k of them are equal, or (2) some k k of them are unequal. We show that the minimum depth of a linear decision tree for these problems is at least (1) max { n − 1 , n lo g 3 ( n / 3 k ) } {\text {max}}\{ n - 1,\quad n\;{\text {lo}}{{\text {g}}_3}(n/3k)\} , and (2) max { n − 1 , n lo g 3 ( k − 1 ) − k + 1 } {\text {max}}\{ n - 1,\quad n\;{\text {lo}}{{\text {g}}_3}(k - 1) - k + 1\} . Our main lower bound for the size of linear decision trees for polyhedra P P in R n {{\mathbf {R}}^n} is given by the sum of Betti numbers for the complement R n ∖ P {{\mathbf {R}}^n}\backslash P . The applications of this general topological bound involve the computation of the Möbius function of intersection lattices of certain subspace arrangements. In particular, this leads to computing various expressions for the Möbius function of posets of partitions with restricted block sizes. Some of these formulas have topological meaning. For instance, we derive a formula for the Euler characteristic of the subset of R n {{\mathbf {R}}^n} of points with no k k coordinates equal in terms of the roots of the truncated exponential ∑ i > k x i / i ! \sum \nolimits _{i > k} {{x^i}} /i! .

Umbral interpolation and the addition/contraction tree for graphs

Given a finite graph G it is possible to define an umbral chromatic polynomial χø(G;x) using the poset of admissible partitions of the vertices of G. We show here that an alternative construction is possible, by developing an umbral interpolation formula which generalises the classical Newton forward interpolation formula. We also explain how the new approach relates directly to the addition/contraction tree of G, and results in a simple algorithmic procedure for computing χø(G;x).

Edges in the poset of partitions of an integer

The edges in the Hasse diagram of the partitions of an integer ordered by reverse refinement are studied. A recursive count of the edges between ranks is developed.

On the poset of partitions of an integer

We study the posets (partially ordered sets) P n of partitions of an integer n, ordered by refinement, as defined by G. Birkhoff, “Lattice Theory” (3rd ed.) Colloq. Publ. Vol. 25, 1967, Amer. Math. Soc. Providince, R.I. In particular we disprove the conjecture that the posets P n are Cohen-Macaulay for all n, and show that even the Möbius functions on the intervals does not alternate in sign in general.