7 problems

Let $m$ and $n$ be grid side lengths, and consider partitions of the $m\times n$ grid into connected pieces of size $3$, called 3-omino partitions. A partition is locked if no reco…

A locked $t$-omino partition is a partition of a grid into connected pieces of size $t$ such that no recombination move can produce a different partition. Let $t$ range over the po…

Let $G(n,n)$ be the $n\times n$ grid graph, and let $\mathcal{M}(G(n,n),3)$ be the metagraph whose vertices are partitions of $G(n,n)$ into three connected subgraphs of equal size,…

Conjecture 4. For all $k\geq 0$, the residue of $L_0(m)$ modulo $2^k$ is periodic with period dividing $2^k$; likewise, the residue of $L_1(m)$ modulo $2^k$ has period dividing…

Let $L(n)$, for $n\geq 0$, be the number of tilings of the Aztec diamond of order $n$ using horizontal skew tetrominos and square tetrominos. Write the multiplicity of 2 in the pri…

Let $M(n)$, for $n\geq 0$, be the number of tilings of the Aztec diamond of order $n$ using dominos and square tetrominos. Conjecture 2. For all $k\geq 1$, if … then … This is pres…

Let $M(n)$, for $n\geq 0$, be the number of tilings of the Aztec diamond of order $n$ using dominos and square tetrominos. Conjecture 1. For all $k\geq 1$, the residue of $M(n)$ mo…