W. H. Mills

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.

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.

On the covering of pairs by quadruples. II

Abstract Let S be a finite set of order n. Let C(n, 4, 2) be the minimum number of quadruples such that each pair of elements of S is contained in at least one of them. In this paper C(n, 4, 2) is determined for all n divisible by 3. The case in which n is not divisible by 3 will be treated in a second paper.

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.

Covering triples by quadruples: an asymptotic solution

Let C(3, 4, n) be the minimum number of four-element subsets (called blocks) of an n-element set, X, such that each three-element subset of X is contained in at least one block. Let L(3, 4, n) = ⌜n4⌜n−13⌜n−22⌝⌝⌝. Schoenheim has shown that C(3, 4, n) ⩾ L(3, 4, n). The construction of Steiner quadruple systems of all orders n≡2 or 4 (mod 6) by Hanani (Canad. J. Math. 12 (1960), 145–157) can be used to show that C(3, 4, n) = L(3, 4, n) for all n ≡ 2, 3, 4 or 5(od 6) and all n ≡ 1 (mod 12). The case n ≡ 7 (mod 12) is made more difficult by the fact that C(3, 4, 7) = L(3, 4, 7) + 1 and until recently no other value for C(3, 4, n) with n≡7 (mod 12) was known. In 1980 Mills showed by construction that C(3, 4, 499) = L(3, 4, 499). We use this construction and some recursive techniques to show that C(3, 4, n) = L(3, 4, n) for all n ⩾ 52423. We also show that if C(3, 4, n) = L(3, 4, n) for n = 31, 43, 55 and if a certain configuration on 54 points exists then C(3, 4, n) = L(3, 4, n) for all n ≠ 7 with the possible exceptions of n = 19 and n = 67. If we assume only C(3, 4, n) = L(3, 4, n) for n = 31 and 43 we can deduce that C(3, 4, n) = L(3, 4, n) for all n ≠ 7 with the possible exceptions of n ϵ {19, 55, 67, 173, 487}.

The Existence of Kirkman Squares—Doubly Resolvable ( v ,3,1)- BIBD s

AbstractA Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v;μ,λ), is a t×t array (t=λ(v−1)/μ(k−1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k,λ)-BIBD. For μ=1, the existence of a KSk(v; μ, λ) is equivalent to the existence of a doubly resolvable (v, k, λ)-BIBD. The spectrum of KS2 (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with λ=1. We show that there exist KS3 (v; 1, 1) for

Covering pairs by quintuples: the case v congruent to 3 (mod 4)

v \equiv 3{\text{ (mod 6)}}

On l-covers of pairs by quintuples: v odd

, v=3 and v≥27 with at most 23 possible exceptions for v.