An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1 + 1/4 + 1/9 + · · · + 1/k 2 conjectured by G. Parisi for the case k = m = n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev [V1, V2, V3]. In this paper we study the complexes of not i-connected k-hypergraphs on n vertices. We show that the complex of not 2-connected graphs has the homotopy type of a wedge of (n − 2)! spheres of dimension 2n − 5. This answers one of the questions raised by Vassiliev [V3] in connection with knot invariants. For this case the S naction on the homology of the complex is also determined. For complexes of not 2-connected k-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n − 2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n − 3)-connected graphs we provide a formula for the generating function of the Euler characteristic.1 There is Z 3 -torsion of rank 1. No Z p -torsion for p = 2, 5 ≤ p ≤ 17. 2 There is Z 3 -torsion of rank 8. No Z p -torsion for p = 2, 5 ≤ p ≤ 17. 3 There is Z 3 -torsion of rank 1. No Z p -torsion for p = 2, 5, 7. 4 There is Z 3 -torsion of rank 35. No Z p -torsion for p = 2, 5, 7. 5 There is Z 3 -torsion of rank 56. No Z p -torsion for p = 2, 5, 7.

The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity −1. This conjecture was recently proved by Jonsson.Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the cylindric case, by some power of 2.We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres.Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity −1. These

Abstract. We reinterpret and generalize conjectures of Lam and Williams as statements about the stationary distribution of a multispecies exclusion process on the ring. The central objects in our study are the multiline queues of Ferrari and Martin. We make some progress on some of the conjectures in different directions. First, we prove Lam and Williams' conjectures in two special cases by generalizing the rates of the Ferrari-Martin transitions. Secondly, we define a new process on multiline queues, which have a certain minimality property. This gives another proof for one of the special cases; namely arbitrary jump rates for three species.

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