Friends and Strangers Walking on Graphs

Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, we define a graph $\mathsf{FS}(X,Y)$ whose vertex set consists of all bijections $\sigma:V(X)\to V(Y)$, where two bijections $\sigma$ and $\sigma'$ are adjacent if they agree everywhere except for two adjacent vertices $a,b \in V(X)$ such that $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y$. This setup, which has a natural interpretation in terms of friends and strangers walking on graphs, provides a common generalization of Cayley graphs of symmetric groups generated by transpositions, the famous $15$-puzzle, generalizations of the $15$-puzzle as studied by Wilson, and work of Stanley related to flag $h$-vectors. We derive several general results about the graphs $\mathsf{FS}(X,Y)$ before focusing our attention on some specific choices of $X$. When $X$ is a path graph, we show that the connected components of $\mathsf{FS}(X,Y)$ correspond to the acyclic orientations of the complement of $Y$. When $X$ is a cycle, we obtain a full description of the connected components of $\mathsf{FS}(X,Y)$ in terms of toric acyclic orientations of the complement of $Y$. We then derive various necessary and/or sufficient conditions on the graphs $X$ and $Y$ that guarantee the connectedness of $\mathsf{FS}(X,Y)$. Finally, we raise several promising further questions.Comment: 28 pages, 5 figure

The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes

Abstract Gelfand–Tsetlin polytopes arise in representation theory and algebraic combinatorics. One can construct the Gelfand–Tsetlin polytope $$\mathrm{GT}_\lambda$$ GT λ for any partition $$\lambda = (\lambda _1,\ldots ,\lambda _n)$$ λ = ( λ 1 , … , λ n ) of weakly increasing positive integers. The integral points in a Gelfand–Tsetlin polytope are in bijection with semi-standard Young tableau of shape $$\lambda$$ λ and parametrize a basis of the $$\mathrm{GL}_n$$ GL n -module with highest weight $$\lambda$$ λ . The combinatorial geometry of Gelfand–Tsetlin polytopes has been of recent interest. Researchers have created new combinatorial models for the integral points and studied the enumeration of the vertices of these polytopes. In this paper, we determine the exact formulas for the diameter of the 1-skeleton, $$\mathrm{diam}(\mathrm{GT}_\lambda )$$ diam ( GT λ ) , and the combinatorial automorphism group, $$\mathrm{Aut}(\mathrm{GT}_\lambda )$$ Aut ( GT λ ) , of any Gelfand–Tsetlin polytope. We exhibit two vertices that are separated by at least $$\mathrm{diam}(\mathrm{GT}_\lambda )$$ diam ( GT λ ) edges and provide an algorithm to construct a path of length at most $$\mathrm{diam}(\mathrm{GT}_\lambda )$$ diam ( GT λ ) between any two vertices. To identify the automorphism group, we study $$\mathrm{GT}_\lambda$$ GT λ using combinatorial objects called $$ladder diagrams$$ ladderdiagrams and examine faces of co-dimension 2