David P. Robbins

Alternating sign matrices and descending plane partitions

Abstract An alternating sign matrix is a square matrix such that (i) all entries are 1, −1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math. 53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math. 66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.

Determinants and alternating sign matrices

Let M be an n by n matrix. By a connected minor of M of size k we mean a minor formed from k consecutive rows and k consecutive columns. We give formulas for det M in terms of connected minors, one involving minors of two consecutive sizes and one involving minors of three consecutive sizes. The formulas express det M as sums indexed by sets of alternating sign matrices. These matrices are described here and by W. H. Mills, D. P. Robbins, and H. Rumsey, Jr. (Invent. Math.66 (1982), 73–87; J. Combin. Theory Ser. A.34 (1983), 340–359). The former study has led to the solution of Macdonalds conjecture on cyclically symmetric plane partitions (G. E. Andrews, Invent. Math.53 (1979), 193–225; I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” p. 53, Oxford Univ. Press (Clarendon), Oxford, 1979).

Genus distributions for bouquets of circles

Abstract The genus distribution of a graph G is defined to be the sequence {gm} such that gm is the number of different imbeddings of G in the closed orientable surface of genus m. A counting formula of D. M. Jackson concerning the cycle structure of permutations is used to derive the genus distribution for any bouquet of circles Bn. It is proved that all these genus distributions for bouquets are strongly unimodal.

Continued fractions for certain algebraic power series

Abstract Let F be an arbitrary field and let K = F (( x −1 )) be the field of formal Laurent series in x −1 over F . The usual theory of continued fractions carries over to K , with the polynomials in x playing the role of the integers. We study the continued fraction expansions of elements of K which are algebraic over F ( x ), the field of rational functions of x . We give the first explicit expansions of algebraic elements of degree greater than 2 for which the degrees of the partial quotients are bounded. In particular we give explicitly the continued fraction expansion for the solution f in K of the cubic equation xf 3 + f + x = 0 when F = GF (2). This cubic was studied by Baum and Sweet. We give examples, for every field F of characteristic greater than 2, of algebraic elements of degree greater than 2 whose partial quotients are all linear, and we give these expansions explicitly. These are the first known examples with partial quotients of bounded degree when F has characteristic greater than 2.

Self-complementary totally symmetric plane partitions

Abstract A totally symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box X n = [1, n ] × [1, n ] × [1, n ] and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the box X n that do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex ( n + 1, n + 1, n + 1). A totally symmetric plane partition is self-complementary if it is congruent (in the geometrical sense) to its complement. This cannot occur unless n = 2 m is even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship with m by m alternating sign matrices. In an earlier paper we described the close connection between m by m alternating sign matrices and descending plane partitions with no parts exceeding m . We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.

Enumeration of a symmetry class of plane partitions

Abstract A cyclically symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box n =[1, n ]×[1, n ]×[1, n ] and which is mapped to itself by cyclic permutations of the coordinate axes. Given a cyclically symmetric plane partition with Ferrers graph F , we can form its transpose-complement, the plane partition whose Ferrers graph is the set of all triples ( i , j , k ) such that ( n + 1 − j , n + 1 − i , n + 1 − k ) ∋ F . This is again a cyclically symmetric plane partition. A cyclically symmetric plane partition is tc-symmetric if it is equal to its transpose-complement. THis cannot occur unless n is even. In this paper we show that the number of tc -symmetric plane partitions in H 2n is given by π k=0 n−1 (3K+1)(6k)!(2k)! (4k+1)!(4k)! We show that, with a suitable assignment of weights, the generating function for tc -symmetric plane partitions divides the generating function for all cyclically symmetric plane partitions. We give analogous results for other classes of plane partitions including descending plane partitions.

On the volume of the polytope of doubly stochastic matrices

We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one). We describe two methods. The first involves a decompos ition of the polytope into simplices. The second involves the enumeration of “magic squares”, that is, n × n nonnegative integer matrices whose rows and columns all sum to the same integer. We have used the first method to confirm the previously known values through n = 7. This method can also be used to compute the volumes of faces of Bn For example, we have observed that the volume of a particular face of Bn appears to be a product of Catalan numbers. We have used the second method to find the volume for n = 8, which we believe was not previously known.

Areas of polygons inscribed in a circle

AbstractHeron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9vqFf0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVe0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2% da9maakaaabaGaam4CaiaacIcacaWGZbGaeyOeI0IaamyyaiaacMca% caGGOaGaam4CaiabgkHiTiaadkgacaGGPaGaaiikaiaadohacqGHsi% slcaWGJbGaaiykaaWcbeaakiaacYcaaaa!4935!

The asymptotic probability that a random biased matrix is invertible

K = \sqrt {s(s - a)(s - b)(s - c)} ,