Albert Nijenhuis

A probabilistic proof of a formula for the number of Young tableaux of a given shape

Publisher Summary This chapter presents a probabilistic proof of a formula for the number of Young tableaux of a given shape. A Young tableau of shape λ is an arrangement of the integers 1, 2,…, n in the cells of the Ferrers diagram of λ such that all rows and columns form increasing sequences. The chapter also presents the problem of the occurrence of hook lengths, which do not seem to be involved naturally in any direct combinatorial correspondence. In any standard tableau, the integer n must appear at a corner, that is, a cell that is at the end of some row and at the end of a column.

Representations of integers by linear forms in nonnegative integers

Abstract Let Ω be the set of positive integers that are omitted values of the form f = Σi=1naixi, where the ai are fixed and relatively prime natural numbers and the xi are variable nonnegative integers. Set ω = #Ω and κ = max Ω + 1 (the conductor). Properties of ω and κ are studied, such as an estimate for ω (similar to one found by Brauer) and the inequality 2ω ≥ κ. The so-called Gorenstein condition is shown to be equivalent to 2ω = κ.

The enumeration of connected graphs and linked diagrams

Abstract Constructive combinatorial proofs are given for recurrence formulas which count, respectively, labeled connected graphs and linked diagrams. Previous proofs were analytical in nature. Our combinatorial proofs give rise to algorithms for selecting at random a labeled connected graph or a linked diagram.

Solving Kepler's equation with high efficiency and accuracy

We present a method for solving Keplers equation for elliptical orbits that represents a gain in efficiency and accuracy compared with those currently in use. The gain is obtained through a starter algorithm which uses Mikkolas ideas in a critical range, and less costly methods elsewhere. A higher-order Newton method is used thereafter. Our method requires two trigonometric evaluations.

Bijective methods in the theory of finite vector spaces

Combinatorial properties of vector spaces over finite fields have been extensively investigated (see Goldman and Rota [ 1, 21, Knuth [3], Milne [4], Calabi and Wilf [S], etc.). In this paper we will obtain a number of results by a unified method. The method, as used in [5], is the observation that the canonical invariant of a vector subspace over a finite field is a matrix over the field, in reduced row echelon form (rref), whose rows span the subspace. If two such matrices differ in even a single entry then they represent different vector subspaces. Combinatorially this means that to count subspaces we just count matrices in rref. Here are the results we obtain in this way:

On permanents and the zeros of rook polynomials

Abstract The concept of rook polynomial of a “chessboard” may be generalized to the rook polynomial of an arbitrary rectangular matrix. A conjecture that the rook polynomials of “chessboards” have only real zeros is thus carried over to the rook polynomials of nonnegative matrices. This paper proves these conjectures, and establishes interlacing properties for the zeros of the rook polynomials of a positive matrix and the matrix obtained by striking any one row or any one column.

Periodicities of Partition Functions and Stirling Numbers modulo p

If p(n, k) is the number of partitions of n into parts ≤k, then the sequence {p(k, k), p(k + 1, k),…} is periodic modulo a prime p. We find the minimum period Q = Q(k, p) of this sequence. More generally, we find the minimum period, modulo p, of {p(n; T)}n ≥ 0, the number of partitions of n whose parts all lie in a fixed finite set T of positive integers. We find the minimum period, modulo p, of {S(k, k), S(k + 1, k),…}, where these are the Stirling numbers of the second kind. Some related congruences are proved. The methods involve the use of cyclotomic polynomials over Zp[x].

The Backtrack Method (BACKTR)

This chapter presents an algorithm BACKTR focusing on the backtrack method. The backtrack method is a reasonable approach to use on problems of exhaustive search when all possibilities must be enumerated or processed. The backtrack procedure grows the vector from left to right and tests at each stage to see if the partially constructed vector has any chance to be extended to a vector which satisfies ζ. If not, the partial vector is rejected, then moving on to the next one, thereby saving the effort of constructing the descendants of a clearly unsuitable partial vector. The chapter discusses the computer implementation of this procedure. The aim is to split off the universal aspects of the backtrack method as a subroutine, which will be useful in most, or all, applications, and to leave the part of the application that differs from one situation to the next to the user, as a program which the user must prepare within certain guidelines. In this approach, it is supposed that the user wishes to prepare a program that will exhibit one vector at a time, satisfying his condition ζ.

Random Permutation of n Letters (RANPER)

This chapter describes an algorithm for generating random permutation of n letters. The random permutation is generated by a sequence of random interchanges. First any one of the n letters 1, …, n for a 1 is chosen. Then, any one of the remaining letters for a 2 is chosen and so on. The construction of a permutation, thus, involves n choices, with respective probabilities and the probability of a given permutation chosen being 1/ n ! The FORTRAN program of algorithm RANPER contains a LOGICAL parameter SETUP. If it is set. FALSE., the subprogram will not setup the array A with 1, …, n but, instead, will operate on whatever data the user has supplied. The chapter presents the sample output for the subroutine. In the analysis presented in the chapter, for each n = 3,……,8, a set of 50 random permutations of n letters was chosen and the number of cycles of each of these 50 permutations was found.

Trace-free differential invariants of triples of vector 1-forms

Abstract It is shown that the “trace-free” differential invariants of triples of vector 1-forms form a space of dimension 13. Twelve of these are accounted for by constructions based on the known bilinear “bracket” of vector 1-forms. We find one that is new, and exhibit it in various forms, including one that shows an unusual symmetry: it alternates in the three vector 1-forms and is a tensor of type (1,2), symmetric in its covariant part. Two-dimensional manifolds admit yet another new invariant.