581 research outputs found
Determinants of incidence and Hessian matrices arising from the vector space lattice
Let be the lattice of subspaces
of the -dimensional vector space over the finite field and
let be the graded Gorenstein algebra defined over
which has as a basis. Let be the Macaulay dual
generator for . We compute explicitly the Hessian determinant
evaluated at the point and relate it to the determinant of the incidence
matrix between and . Our exploration is
motivated by the fact that both of these matrices arise naturally in the study
of the Sperner property of the lattice and the Lefschetz property for the
graded Artinian Gorenstein algebra associated to it
Lower Bounds for Real Solutions to Sparse Polynomial Systems
We show how to construct sparse polynomial systems that have non-trivial
lower bounds on their numbers of real solutions. These are unmixed systems
associated to certain polytopes. For the order polytope of a poset P this lower
bound is the sign-imbalance of P and it holds if all maximal chains of P have
length of the same parity. This theory also gives lower bounds in the real
Schubert calculus through sagbi degeneration of the Grassmannian to a toric
variety, and thus recovers a result of Eremenko and Gabrielov.Comment: 31 pages. Minor revision
Face module for realizable Z-matroids
In this work, we define the face module for a realizable matroid over Z. Its
Hilbert series is, indeed, the expected specialization of the Grothendieck -
Tutte polynomial defined by Fink and Moci.
This work will appear in 'Contributions to Discrete Mathematics'Comment: 16 pages, 4 figures, 1 Tabl
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
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
Paths to Understanding Birational Rowmotion on Products of Two Chains
Birational rowmotion is an action on the space of assignments of rational
functions to the elements of a finite partially-ordered set (poset). It is
lifted from the well-studied rowmotion map on order ideals (equivariantly on
antichains) of a poset , which when iterated on special posets, has
unexpectedly nice properties in terms of periodicity, cyclic sieving, and
homomesy (statistics whose averages over each orbit are constant) [AST11, BW74,
CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context,
rowmotion appears to be related to Auslander-Reiten translation on certain
quivers, and birational rowmotion to -systems of type
described in Zamolodchikov periodicity.
We give a formula in terms of families of non-intersecting lattice paths for
iterated actions of the birational rowmotion map on a product of two chains.
This allows us to give a much simpler direct proof of the key fact that the
period of this map on a product of chains of lengths and is
(first proved by D.~Grinberg and the second author), as well as the first proof
of the birational analogue of homomesy along files for such posets.Comment: 31 pages, to appear in Algebraic Combinatoric
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