combinatorics

I am interested in finding a general formula for the total number of regions created by an iterative process of drawing nested polygons inside a regular $n$-gon by joining midpoints. Process ...

From a corner-on view of a Rubik's Cube, calculate the number of solvable arrangements of the cube. Your input will provide the state of a Rubik's cube when viewed like the image below: The input ...

Let $ U(\mathbb{R}^2) $ be the unit distance graph of the plane (vertices = points in $\mathbb{R}^2$, edges = pairs at Euclidean distance exactly 1). The chromatic number $\chi(U)$ is known to be ...

Is it possible to place a king (K), a rook (R), 2 knights (N), and 2 bishops (B) into the following chess grid of 12 squares so that no piece attacks any other piece? If yes, how many ways are there?

Given $n$ distinct points in the Euclidean plane, what is the greatest number of pairs of points that can be unit distance apart? Paul Erdős conjectured that the answer was $n^{1+o(1)}$. Recently, ...

Fix integers and and set . Let denote the complete -partite -uniform hypergraph with parts of size . We prove that the Zarankiewicz number provided . Previously this was known only for due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of , for example, it gives where the exponent 11/4 is …

Is there a formal definition for the concept of combinatorial interpretation? I have read in different places that a sequence of non-negative integers has a combinatorial interpretation if the members ...

See the same problem but for T-tetrominoes. Consider the infinite square grid whose squares are each colored black or white. What is the minimum required density of cells to be colored black such that ...

Given an alphabet of $p$ letters, how can I find a minimal set $F$ of words of length at most $p$ such that every two-letters word appears as a subsequence (subword) in exactly one of the words in ...

Easy Random Trees Can you think of a way to efficiently generate a random plane tree? Richard P. Stanley in his book Catalan Numbers has a really nifty combinatorial proof of why Catalan numbers have the formula \[ C_n = {1 \over n+1}{2n \choose n} \] The standard proof uses generating functions applied to an inductive definition of the Catalan numbers, which frankly does little to illumiate thei…

In this paper we present enumerative results for Stirling numbers of the first kind for two graph products, the matched product and the m-star, using the combinatorial model of rearrangements. The kth Stirling number of the first kind for a simple graph G counts the number of ways to decompose G into exactly k vertex-disjoint cycles, including single vertices as 1-cycles, single edges as 2-cycles…

Given 20 distinct positive integers not greater than 99. Show that among their pairwise differences, at least four are equal, or find the counterexample. So the problem is simliar to Given 20 distinct ...

What is the maximal number of knights on a square chessboard which can be placed in such a way that every knight attacks exactly two others? We name $f(n)$ that maximal number of them on a $n \times ...

Let ${n \brack k}$ be the unsigned Stirling numbers of the first kind. With the luck of intuition and after a lot of numerical experiments I conjecture that $$ \sum\limits_{k=0}^{n} ...

A Langford pairing is an arrangement of $2n$ numbers, $1, 1, 2, 2, \ldots, n, n$, in a row such that there are $k$ numbers between the two $k$s for any $k=1,2,\ldots,n$. These are well studied and are ...

ACO Seminar - Zion Hefty Wean Hall 8220 Anonymous (not verified) Mon, 03/23/2026 - 10:19 In Person Improving R(3,k) in just two bites ZION HEFTY The Ramsey number R(t,k) is the smallest n such that any red-blue edge coloring of the n-vertex complete graph has either a t-vertex red complete subgraph or a k-vertex blue complete subgraph. We will investigate the…

ACO Seminar - Tracy Chin Wean Hall 8220 Anonymous (not verified) Mon, 03/16/2026 - 09:37 In Person Valuated Delta Matroids and Principal Minors TRACY CHIN Delta matroids are a generalization of matroids that arise naturally from combinatorial objects such as matchings, ribbon graphs, and principal minors of symmetric and skew symmetric matrices. In this talk,…

ACO Seminar - Daniel Zhu Wean 6220 Anonymous (not verified) Mon, 03/09/2026 - 16:30 In Person Schur complements for tensors and multilinear commutative rank DANIEL ZHU We show that three notions of ranks for matrices of multilinear forms are equivalent. This result generalizes a classical result of Flanders, corrects a minor hole in work of Fortin and Reutena…

Theory Lunch Seminar - Zeyu Zheng Gates Hillman 8102 Anonymous (not verified) Tue, 02/24/2026 - 10:44 In Person The generalized trifference problem ZEYU ZHENG We study the problem of finding the largest number T(n,m) of ternary vectors of length n such that for any three distinct vectors there are at least m coordinates where they pairwise differ. For m=1 , t…

ACO Seminar - Carl Schildkraut Wean Hall 8220 Anonymous (not verified) Mon, 02/23/2026 - 10:38 In Person Ramsey Cayley graphs over non-abelian groups CARL SCHILDKRAUT A conjecture of Alon states that, for some absolute constant C, every finite group G possesses a Cayley graph with clique and independence number each at most C*log|G|. Recently, Conlon, Fox, Ph…