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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
If you are the smartest person in the room, you are in th...
Davide Sclosa · 2024-07-11 · via math.CO updates on arXiv.org

If taken seriously, the advice in the title leads to interesting combinatorics. Consider $N$ people moving between $M$ rooms as follows: at each step, simultaneously, the smartest person in each room moves to a different room of their choice, while no one else moves. The process repeats. In this paper we determine which configurations are reachable, from which other configurations, and provide bounds on the number of moves. Namely, let $G(N,M)$ be the directed graph with vertices representing all $M^N$ configurations and edges representing possible moves. We prove that the graph $G(N,M)$ is weakly connected, and that it is strongly connected if and only if $M\geq N+1$ (one extra room for maneuvering is both required and sufficient). For $M\leq N$, we show that the graph has a giant strongly connected component with $Θ(M^N)$ vertices and diameter $\mathcal O(N^2)$.