On ideals with the Rees property

A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S$ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal $I \subset S$ is said to be $\mathfrak m$-full if $\mathfrak mI:y=I$ for some $y \in \mathfrak m$, where $\mathfrak m$ is the graded maximal ideal of $S$. It was proved by one of the authors that $\mathfrak m$-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not $\mathfrak m$-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.Comment: 8 page

On the Sperner property for the absolute order on complex reflection groups

Two partial orders on a reflection group, the codimension order and the prefix order, are together called the absolute order when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type $D_n$, for which this property is conjectural. The Sperner property had previously been established for the noncrossing partition lattice $NC_W$, a certain maximal interval in the absolute order, but not for the entire poset, except in the case of the symmetric group. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.Comment: 12 pages, comments welcome; v2: minor edits and journal referenc

A finite word poset : In honor of Aviezri Fraenkel on the occasion of his 70th birthday

Our word posets have �nite words of bounded length as their elements, with the words composed from a �nite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and R¨ohl [4]) and a BLYM inequality is veri�ed (via the normalized matching property)

Gibbs distributions for random partitions generated by a fragmentation process

In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w\_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent process with affine collision rate K\_{i,j}=a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton-Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions.Comment: 38 pages, 2 figures, version considerably modified. To appear in the Journal of Statistical Physic

Shelling the m=1 amplituhedron

The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a projection inside the Grassmannian $Gr_{k,k+m}$ of the totally nonnegative part of $Gr_{k,n}$. Karp and Williams (2019) studied the $m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of it. Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective sign vectors of length $n$ with exactly $l$ sign changes. We show that $R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that $\mathcal{A}_{n,k,1}$ is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset $P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign vectors of length $n$ with at most $l$ sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.Comment: 19 page

Negative correlation and log-concavity

We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea et al.; prove that "almost exchangeable" measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions.Comment: 21 pages; only minor changes since previous version; accepted for publication in Random Structures and Algorithm

Dedekind's problem in the hypergrid

Consider the partially ordered set on $[t]^n:=\{0,\dots,t-1\}^n$ equipped with the natural coordinate-wise ordering. Let $A(t,n)$ denote the number of antichains of this poset. The quantity $A(t,n)$ has a number of combinatorial interpretations: it is precisely the number of $(n-1)$-dimensional partitions with entries from $\{0,\dots,t\}$, and by a result of Moshkovitz and Shapira, $A(t,n)+1$ is equal to the $n$-color Ramsey number of monotone paths of length $t$ in 3-uniform hypergraphs. This has led to significant interest in the growth rate of $A(t,n)$. A number of results in the literature show that $\log_2 A(t,n)=(1+o(1))\cdot \alpha(t,n)$, where $\alpha(t,n)$ is the width of $[t]^n$, and the $o(1)$ term goes to $0$ for $t$ fixed and $n$ tending to infinity. In the present paper, we prove the first bound that is close to optimal in the case where $t$ is arbitrarily large compared to $n$, as well as improve all previous results for sufficiently large $n$. In particular, we prove that there is an absolute constant $c$ such that for every $t,n\geq 2$, $$\log_2 A(t,n)\leq \left(1+c\cdot \frac{(\log n)^3}{n}\right)\cdot \alpha(t,n).$$ This resolves a conjecture of Moshkovitz and Shapira. A key ingredient in our proof is the construction of a normalized matching flow on the cover graph of the poset $[t]^n$ in which the distribution of weights is close to uniform, a result that may be of independent interest.Comment: 28 pages + 4 page Appendix, 3 figure

A Characterization of LYM and Rank Logarithmically Concave Partially Ordered Sets and Its Applications

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem