22,451 research outputs found
Stability of Kronecker coefficients via discrete tomography
In this paper we give a new sufficient condition for a general stability of
Kronecker coefficients, which we call it additive stability. It was motivated
by a recent talk of J. Stembridge at the conference in honor of Richard P.
Stanley's 70th birthday, and it is based on work of the author on discrete
tomography along the years. The main contribution of this paper is the
discovery of the connection between additivity of integer matrices and
stability of Kronecker coefficients. Additivity, in our context, is a concept
from discrete tomography. Its advantage is that it is very easy to produce lots
of examples of additive matrices and therefore of new instances of stability
properties. We also show that Stembridge's hypothesis and additivity are
closely related, and prove that all stability properties of Kronecker
coefficients discovered before fit into additive stability.Comment: 22 page
An integration of Euler's pentagonal partition
A recurrent formula is presented, for the enumeration of the compositions of
positive integers as sums over multisets of positive integers, that closely
resembles Euler's recurrence based on the pentagonal numbers, but where the
coefficients result from a discrete integration of Euler's coefficients. Both a
bijective proof and one based on generating functions show the equivalence of
the subject recurrences.Comment: 22 pages, 2 figures. The recurrence investigated in this paper is
essentially that proposed in Exercise 5.2.3 of Igor Pak's "Partition
bijections, a survey", Ramanujan J. 12 (2006), but casted in a different form
and, perhaps more interestingly, endowed with a bijective proof which arises
from a construction by induction on maximal part
A polytope related to empirical distributions, plane trees, parking functions, and the associahedron
We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0,
whose combinatorial properties are closely connected with empirical
distributions, plane trees, plane partitions, parking functions, and the
associahedron. In particular, we give explicit formulas for the volume of
Pi_n(x) and, when the x_i's are integers, the number of integer points in
Pi_n(x). We give two polyhedral decompositions of Pi_n(x), one related to order
cones of posets and the other to the associahedron.Comment: 41 page
Promotion and Rowmotion
We present an equivariant bijection between two actions--promotion and
rowmotion--on order ideals in certain posets. This bijection simultaneously
generalizes a result of R. Stanley concerning promotion on the linear
extensions of two disjoint chains and recent work of D. Armstrong, C. Stump,
and H. Thomas on root posets and noncrossing partitions. We apply this
bijection to several classes of posets, obtaining equivariant bijections to
various known objects under rotation. We extend the same idea to give an
equivariant bijection between alternating sign matrices under rowmotion and
under B. Wieland's gyration. Finally, we define two actions with related orders
on alternating sign matrices and totally symmetric self-complementary plane
partitions.Comment: 25 pages, 22 figures; final versio
The shape of random tanglegrams
A tanglegram consists of two binary rooted trees with the same number of
leaves and a perfect matching between the leaves of the trees. We show that the
two halves of a random tanglegram essentially look like two independently
chosen random plane binary trees. This fact is used to derive a number of
results on the shape of random tanglegrams, including theorems on the number of
cherries and generally occurrences of subtrees, the root branches, the number
of automorphisms, and the height. For each of these, we obtain limiting
probabilities or distributions. Finally, we investigate the number of matched
cherries, for which the limiting distribution is identified as well
A second look at the toric h-polynomial of a cubical complex
We provide an explicit formula for the toric -contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the -polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction
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