Another short proof of the Joni-Rota-Godsil integral formula for counting bipartite matchings

How many perfect matchings are contained in a given bipartite graph? An exercise in Godsil's 1993 \textit{Algebraic Combinatorics} solicits proof that this question's answer is an integral involving a certain rook polynomial. Though not widely known, this result appears implicitly in Riordan's 1958 \textit{An Introduction to Combinatorial Analysis}. It was stated more explicitly and proved independently by S.A.~Joni and G.-C.~Rota [\textit{JCTA} \textbf{29} (1980), 59--73] and C.D.~Godsil [\textit{Combinatorica} \textbf{1} (1981), 257--262]. Another generation later, perhaps it's time both to simplify the proof and to broaden the formula's reach

Exchangeable pairs, switchings, and random regular graphs

We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue functionals for random regular graphs. Previous results on the distribution of cycle counts by McKay, Wormald, and Wysocka (2004) used the method of switchings, a combinatorial technique for asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and demonstrates an interesting connection between the two techniques.Comment: Very minor changes; 23 page

A lower bound on the maximum permanent in Λnk

AbstractLet Pnk be the maximum value achieved by the permanent over Λnk, the set of (0,1)-matrices of order n with exactly k ones in each row and column. Brègman proved that Pnk⩽k!n/k. It is shown here that Pnk⩾k!tr! where n=tk+r and 0⩽r<k. From this simple bound we derive that (Pnk)1/n∼k!1/k whenever k=o(n) and deduce a number of structural results about matrices which achieve Pnk. These include restrictions for large n and k on the number of components which may be drawn from Λk+ck for a constant c⩾1.Our results can be directly applied to maximisation problems dealing with the number of extensions to Latin rectangles or the number of perfect matchings in regular bipartite graphs