28 research outputs found
Enumeration of flattened -Stirling permutations with respect to descents
A -Stirling permutation of order is said to be "flattened" if the
leading terms of its increasing runs are in ascending order. We show that
flattened -Stirling permutations of order are in bijection
correspondence with a colored variant of type set partitions of ,
introduced by D.G.L. Wang. Using the theory of weighted labelled structures, we
give the exponential generating functions of their cardinality and their
descent enumerating polynomials. We also provide enumerative formulae for the
number of flattened -Stirling permutations of order with small number of
descents and the number of flattened Stirling permutations with maximum number
of descents.Comment: 22 pages, 1 table. Comments are welcom
Realizing the -permutahedron via flow polytopes
Ceballos and Pons introduced the -weak order on -decreasing trees, for
any weak composition . They proved that it has a lattice structure and
further conjectured that it can be realized as the -skeleton of a polyhedral
subdivision of a polytope. We answer their conjecture in the case where is
a strict composition by providing three geometric realizations of the
-permutahedron. The first one is the dual graph of a triangulation of a flow
polytope of high dimension. The second one, obtained using the Cayley trick, is
the dual graph of a fine mixed subdivision of a sum of hypercubes that has the
conjectured dimension. The third one, obtained using tropical geometry, is the
-skeleton of a polyhedral complex for which we can provide explicit
coordinates of the vertices and whose support is a permutahedron as
conjectured.Comment: 39 pages, 14 figure