Herbert S. Wilf

Mathematical Methods for Digital Computers

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An algorithmic proof theory for hypergeometric (ordinary and q') multisum/integral identities

SummaryIt is shown that every ‘proper-hypergeometric’ multisum/integral identity, orq-identity, with a fixed number of summations and/or integration signs, possesses a short, computer-constructible proof. We give a fast algorithm for finding such proofs. Most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here. We give many examples of the method, including computer-generated proofs of identities of Mehta-Dyson, Selberg, Hille-Hardy,q-Saalschütz, and others. The prospect of using the method for proving multivariate identities that involve an arbitrary number of summations/integrations is discussed.

Recounting the rationals

2. The function values f(n) actually count something nice. In fact, f(n) is the number of ways of writing the integer n as a sum of powers of 2, each power being used at most twice (i.e., once more than the legal limit for binary expansions). For instance, we can write 5 = 4 + 1 = 2 + 2 + 1, so there are two such ways to write 5, and therefore f(5) = 2. Let’s say that f(n) is the number of hyperbinary representations of the integer n.

Mathematics for the physical sciences

Mathematics for the physical sciences , Mathematics for the physical sciences , کتابخانه دیجیتال جندی شاپور اهواز

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.

BACKTRACK: AN O(1) EXPECTED TIME ALGORITHM FOR THE GRAPH COLORING PROBLEM *

Abstract Fix a number K, of colors. We consider the usual backtrack algorithm for the decision problem of K-colorability of a graph G. We show that the algorithm operates in average time that is O(1), as the number of vertices of G approaches infinity. For instance, a backtrack search tree for 3-coloring a graph has an average of about 197 nodes, averaged over all graphs of all sizes.

A unified setting for sequencing, ranking, and selection algorithms for combinatorial objects

The recurrent construction of combinatorial objects proceeds in steps at each of which we modify the partially constructed object by certain elementary operations such as adjunction of a new element, relabeling, etc. The end result of this sequence of decisions is then available, and this subset, permutation, tree, etc. is called the output combinatorial object. It is, of course, just a record of the totality of decisions which were made during its construction, and if it seems profitable to do so, we may regard that sequence of decisions as being itself the combinatorial object, whatever the final presentation may be. Then any natural setting which we construct for dealing with the sequence of decisions itself will ultimately result in procedures for processing the objects. The theory of directed graphs offers such a setting, and from it we are able to present algorithms which deal simultaneously with many families of objects. The prototype of this situation is perhaps the binomial identity

The number of maximal independent sets in a tree

We find the largest number of maximal independent sets of vertices that any tree of n vertices can have.

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 patterns of permutations

Let n,k be positive integers, with k ≤ n, and let τ be a fixed permutation of {1,...,k}. We will call τ the pattern. We will look for the pattern τ in permutations σ of n letters. A pattern τ is said to occur in a permutation σ if there are integers 1 ≤ i1 > i2 > ... > ik ≤ n such that for all 1 ≤ r > s ≤ k we have τ(r) > τ(s) if and only if σ(ir) > σ(is).Example. Suppose τ = (132). Then this pattern of k = 3 letters occurs several times in the following permutation σ, of n = 14 letters (one such occurrence is underlined): σ=(5 2 9 4 14 10 1 3 6 15 8 11 7 13 12).