The Ramanujan conjecture states that the coefficients $\tau(n)$ in the identity $$q\prod_{m=1}^\infty(1-q^m)^{24}=\sum_{n=1}^\infty\tau(n)q^n$$ satisfy the inequality $|\tau(n)|\ …

Here rays are called lines. Call $M(a_1,a_2)$ the number for matrices of length $a_1$ and height $a_2$, made of $0$ and $1$, having neither null vector nor null co-vector. In other …

I am looking at words $\alpha_1 \ldots \alpha_n$, where $\alpha_j \in \{ 1, \ldots, j \}$. Thinking of $\alpha_j$ as a height, these words can be interpreted as left-to-right paths …

This is the sort of problems in combinatorics with a rather innocent look that turn out to be quite challenging - at least for a bunch of physicists! :) Suppose we have a multiset …

In October 2010, I published a Monthly problem that introduced the concept of a fair permutation, which is a permutation $\pi$ such that for every $i$, either $\pi(i) > i$ and $\pi …

Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and …

This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated …

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f …

I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadr …

Consider two different infinite graphs, whose vertices are drawn from $\mathbb Z^2$ or $\mathbb Z^3$. Let $P_d : \mathbb Z^d \times \mathbb N \to \mathbb N$ for $d \in \{2,3\}$ be …