Related papers: de Finetti Lattices and Magog Triangles

The Bourque-Ligh conjecture states that if $S=\{x_1,x_2,\ldots,x_n\}$ is a gcd-closed set of positive integers with distinct elements, then the LCM matrix $[S]=[\hbox{lcm}(x_i,x_j)]$ is invertible. It is well known that this conjecture…

Given integers $p,q,t$ with $1 \le t \le p$ and $1 \le q \le h_p(t)$, a strong $(p,q,t)$-coloring of the Boolean lattice $B_n$ is a coloring of its $t$-chains such that every induced copy of $B_p$ in $B_n$ uses at least $q$ colors on its…

Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions: (1) the size of $P$ is a power…

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to $n \times n$ alternating sign matrices when…

We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of $P^A_n$ and $P^B_n$ are subsets of the symmetric and the hyperoctahedral…

A partial order $\prec$ on $[n]$ is naturally labelled (NL) if $x\prec y$ implies $x<y$. We establish a bijection between {3, 2+2}-free NL posets and 12-34-avoiding permutations, determine functional equations satisfied by their generating…

For integers m, n $\ge$ 1, we describe a bijection sending dissections of the (mn + 2)-regular polygon into (m + 2)-sided polygons to a new basis of the quotient of the polynomial algebra in mn variables by an ideal generated by some kind…

We study D\'iaz-Dineen's problem for regular homogeneous vector-valued polynomials. In particular, we prove that if $E^*$ and $F^*$ are lattice isomorphic with at least one having order continuous norm, then $\mathcal{P}^r(^n E; G^*)$ and…

We are interested in finding an explicit bijection between two families of combinatorial objects: Gog and Magog triangles. These two families are particular classes of Gelfand-Tsetlin triangles and are respectively in bijection with…

We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types,…

A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq {\cal F}$ of sets is a copy of a poset $P$ in ${\cal F}$ if there exists a bijection $\phi:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that whenever $x \le_P x'$ holds, then so does…

This paper investigates a novel connection between reductions of companion matrices associated with a symmetric family of certain binomial ideals in the coordinate ring of affine n-space and permutation matrices. Specifically, for fixed…

For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some…

A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the poset Ramsey number $R(P,P')$ be the…

Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper…

It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…

The Fon-Der-Flaass action partitions the order ideals of a poset into disjoint orbits. For a product of two chains, Propp and Roby observed --- across orbits --- the mean cardinality of the order ideals within an orbit to be invariant. That…

Given a morphism from an affine semigroup Q to an arbitrary commutative monoid, it is shown that every fiber possesses an affine stratification: a partition into a finite disjoint union of translates of normal affine semigroups. The proof…

We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are…

We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non…