Generalized Eulerian Numbers and Directed Friends-and-seats Graphs

Abstract

Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly $m$ descents, or, due to the Foata transform, the number of permutations on $[n]$ with exactly $m$ excedances. Friends-and-seats graphs, also known as friends-and-strangers graphs, are a seemingly unrelated recent construction in graph theory. In this paper, we introduce directed friends-and-seats graphs and establish a connection between these graphs and a generalization of the Eulerian numbers. We use this connection to reprove and extend a Worpitzky-like identity on generalized Eulerian numbers.

Figures (5)

• Figure 1: Example friends graph $(X)$ and seats graph $(Y)$, with people $A,B$ and $C$ and seats numbered from $1$ to $3$.
• Figure 2: The resulting friends-and-seats graph. $ABC$ denotes that $A$ sits on chair 1, $B$ sits on chair 2, and $C$ sits on chair 3.
• Figure 3: An example of a directed seat graph $X$ and friends graph $Y$.
• Figure 4: The directed friends-and-seats graph $\mathop{\mathrm{\mathsf{DFS}}}\nolimits(X,Y)$ resulting from the seat graphs.
• Figure 5: The friends-and-seats graph $\mathop{\mathrm{\mathsf{FS}}}\nolimits(X,Y)$ resulting from the graphs described above.

Theorems & Definitions (25)

• Theorem 1
• proof
• Corollary 2
• Theorem 3
• proof
• Corollary 4
• Theorem 5
• proof
• Theorem 6
• proof