518 research outputs found
Obstructions to combinatorial formulas for plethysm
Motivated by questions of Mulmuley and Stanley we investigate
quasi-polynomials arising in formulas for plethysm. We demonstrate, on the
examples of and , that these need not be counting
functions of inhomogeneous polytopes of dimension equal to the degree of the
quasi-polynomial. It follows that these functions are not, in general, counting
functions of lattice points in any scaled convex bodies, even when restricted
to single rays. Our results also apply to special rectangular Kronecker
coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples;
v3: final version as in Electronic Journal of Combinatoric
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
Generic and special constructions of pure O-sequences
It is shown that the h-vectors of Stanley-Reisner rings of three classes of
matroids are pure O-sequences. The classes are (a) matroids that are
truncations of other matroids, or more generally of Cohen-Macaulay complexes,
(b) matroids whose dual is (rank + 2)-partite, and (c) matroids of
Cohen-Macaulay type at most five. Consequences for the computational search for
a counterexample to a conjecture of Stanley are discussed.Comment: 16 pages, v2: various small improvements, accepted by Bulletin of the
London Math. Societ
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