Counting smaller elements in the Tamari and m-Tamari lattices

We introduce new combinatorial objects, the interval- posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear operator that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear operator on each tree T counts the number of trees smaller than T in the Tamari order. Then we show that a similar m + 1-linear operator is also used in the functionnal equation of m-Tamari intervals. We explain how the m-Tamari lattices can be interpreted in terms of m+1-ary trees or a certain class of binary trees. We then use the interval-posets to recover the functional equation of m-Tamari intervals and to prove a generalized formula that counts the number of elements smaller than or equal to a given tree in the m-Tamari lattice.Comment: 46 pages + 3 pages of code appendix, 27 figures. Long version of arXiv:1212.0751. To appear in Journal of Combinatorial Theory, Series

Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.Comment: 51 page

Random structures for partially ordered sets

This thesis is presented in two parts. In the first part, we study a family of models of random partial orders, called classical sequential growth models, introduced by Rideout and Sorkin as possible models of discrete space-time. We analyse a particular model, called a random binary growth model, and show that the random partial order produced by this model almost surely has infinite dimension. We also give estimates on the size of the largest vertex incomparable to a particular element of the partial order. We show that there is some positive probability that the random partial order does not contain a particular subposet. This contrasts with other existing models of partial orders. We also study "continuum limits" of sequences of classical sequential growth models. We prove results on the structure of these limits when they exist, highlighting a deficiency of these models as models of space-time. In the second part of the thesis, we prove some correlation inequalities for mappings of rooted trees into complete trees. For T a rooted tree we can define the proportion of the total number of embeddings of T into a complete binary tree that map the root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and Morayne states that, for two binary trees with one a subposet of the other, this proportion is larger for the larger tree. They conjecture that the same is true for two arbitrary trees with one a subposet of the other. We disprove this conjecture by analysing the asymptotics of this proportion for large complete binary trees. We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a correlation inequality which enables us to generalise their result in other directions