Sunflowers in lattices.
Sur le nombre d'éléments des niveaux des produits de chaînes et des treillis permutoèdres
Les produits de chaînes comptent parmi les ensembles (partiellement) ordonnés les plus fréquemment rencontrés. On rappelle, avec des démonstrations en partie nouvelles, divers résultats exacts ou approchés sur les cardinaux de leurs niveaux et sur le nombre de ses niveaux de cardinal maximum. Un plongement avec de bonnes propriétés permet d'appliquer ces résultats aux niveaux du permutoèdre (ordre faible de Bruhat sur les permutations).
The action of the symmetric group on a generalized partition semilattice.
The Bruhat rank of a binary symmetric staircase pattern
In this work we show that the Bruhat rank of a symmetric (0,1)-matrix of order n with a staircase pattern, total support, and containing In, is at most 2. Several other related questions are also discussed. Some illustrative examples are presented.
The cd-index of Bruhat intervals.
The Charney-Davis quantity for certain graded posets.
The fundamental group of balanced simplicial complexes and posets.
The induced subgraph order on unlabelled graphs.
The -colored composition poset.
The Milgram non-operad
C. Berger claimed to have constructed an -operad-structure on the permutohedras, whose associated monad is exactly the Milgram model for the free loop spaces. In this paper I will show that this statement is not correct.
The Möbius function of the consecutive pattern poset.
The monoid of ordered partitions of a natural number.
The order dimension of Bruhat order on infinite Coxeter groups.
The ordering of commutative terms
By a commutative term we mean an element of the free commutative groupoid of infinite rank. For two commutative terms , write if contains a subterm that is a substitution instance of . With respect to this relation, is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has...
The ordinal variety of distributive ordered sets of width two
Tight quotients and double quotients in the Bruhat order.
Transitivity and partial order
In this paper we find a one-to-one correspondence between transitive relations and partial orders. On the basis of this correspondence we deduce the recurrence formula for enumeration of their numbers. We also determine the number of all transitive relations on an arbitrary -element set up to .
Two analogues of a classical sequence.
Two new criteria for comparison in the Bruhat order.
Two results on a partial ordering of finite sequences
In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) for which . The function measures the maximum length of finite sequences over symbols which contain no subsequence of the type . It follows from the result of Hart and Sharir that the containment is a (minimal) obstacle to . We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth. In the second part of the paper we investigate whether...