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We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4). Our derivation is based on a tree interpretation of the various encountered generating functions.
We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the natural approac
We show that the definition of parabolic-like map can be slightly modified, by asking $partial Delta$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.
Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient con
This article contains a characterization of operator systems $cS$ with the property that every positive map $phi:cS rightarrow M_n$ is decomposable, as well as an alternate and a more direct proof of a characterization of decomposable maps due to E. Sto rmer.
In the picture-hanging puzzle we are to hang a picture so that the string loops around $n$ nails and the removal of any nail results in a fall of the picture. We show that the length of a sequence representing an element in the free group with $n$ ge