The distribution of longest run lengths in integer compositions (2009)
We find the generating function for $C(n,k,r)$, the number of compositions of $n$ into $k$ positive parts all of whose runs (contiguous blocks of constant parts) have lengths less than $r$, using...
Counting nondecreasing integer sequences that lie below a barrier (2009)
Pemantle, Robin, Wilf, Herbert S.
Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$....
Carla D. Savage, Ira M. Gessel, Herbert S. Wilf
joint distribution of descent and major index over
On a conjecture of Ira Gessel (2008)
Petkovsek, Marko, Wilf, Herbert S.
Let F(m; n1, n2) denote the number of lattice walks from (0,0) to (n1,n2), always staying in the first quadrant {(n_1,n_2); n1 >= 0, n2 >= 0} and having exactly m steps, each of which belongs to the...
corresponds to a legal arrangement of kings on a 6 × 10 board, namely the following one. (2008)
On a 2m × 2n chessboard, the maximum number of nonattacking kings that can be placed is mn, since each 2×2 cell can have at most one king. Let fm(n) denote the number of ways that mn nonattacking...
A Discontinuity in the Distribution of Fixed Point Sums (2008)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
the electronic journal of combinatorics 10 (2003), #R15 1 The quantity f(n, r), defined as the number of permutations of the set [n] = {1,2,...n} whose fixed points sum to r, shows a sharp...
The method of characteristics, and ‘problem 89 ’ of (2008)
Herbert S. Wilf, To Richard Stanley
We apply the method of characteristics for the solution of pde’s to two combinatorial problems. The first is finding an explicit form for a distribution that arises in bioinformatics. The second is...
Edward A. Bender, Gregory F. Lawler, Robin Pemantle, Herbert S. Wilf
Abstract. Let n = b1 + · · · + bk = b ′ 1 + · · · + b ′ k be a pair of compositions of n into k positive parts. We say this pair is irreducible if there is no positive j < k for which b1...
Mathematics: An Experimental Science (2008)
Albert Einstein once said “You can confirm a theory with experiment, but no path leads from experiment to theory. ” But that was before computers. In mathematical research now, there’s a very...
Séminaire Lotharingien de Combinatoire 52 (2004), Article B50h Irreducible compositions (2008)
Edward A. Bender, Gregory F. Lawler, Robin Pemantle, Herbert S. Wilf
and the first return to the origin of a random walk Abstract. Let n = b1 + · · · + bk = b ′ 1 + · · · + b ′ k
The Redheffer matrix of a partially ordered set (2008)
R. Redheffer described an n × n matrix of 0’s and 1’s the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and...
E. Rodney Canfield, Carla D. Savage, Herbert S. Wilf
For integer partitions λ: n = a1 +... + ak, where a1 ≥ a2 ≥... ≥ ak ≥ 1, we study the sum a1 + a3 +... of the parts of odd index. We show that the average of this sum, over all partitions λ...
Counting pairs of lattice paths by intersections (2007)
Ira Gessel, Wayne Goddard, Herbert S. Wilf, Lily Yen
We count the pairs of walks between diagonally opposite corners of a given lattice rectangle by the number of points in which they intersect. We note that the number of such pairs with one...
Dedicated to the memory of Gian-Carlo Rota (2007)
Neil J. Calkin, Herbert S. Wilf, E. Rodney Canfield
We answer a question posed by Lampert and Slater [7]. Consider a sequence of real numbers qn in the interval [0,1] defined by q0 =0,q1 = 1, and, for n ≥ 1, qn+1 equals an average of preceding terms...
Richard Garfield, Herbert S. Wilf
If p is a prime, a is a primitive root modulo p, and n is a positive integer, let r i (n) be the number of k such that 0 k n and \Gamma n
Andrew M. Odlyzko, Herbert S. Wilf
The problem of Josephus is the following. We are given two positive integers n; q. There are n places arranged around a circle, and numbered clockwise 1; 2; : : : ; n. Each
Gert Almkvist, Herbert S. Wilf
The number of partitions of n is given by the asymptotic and convergent series p(n) = 1
We find an interesting relationship between two difficult counting problems in the theory of symmetric functions. For integer k 1, let y k (n) denote the number of standard Young tableaux on n cells...
Lectures on Integer Partitions (2007)
this paper was to prove a Rogers-Ramanujan identity bijectively, and indeed, the authors did prove the identity (and in the process claimed a $100 prize that had been o#ered by George Andrews),...
When Are Subset Sums Equidistributed Modulo M? (2007)
Stan Wagon Macalester, Herbert S. Wilf
. For a triple (n; t; m) of positive integers, we attach to each t-subset S = fa 1 ; : : : ; a t g ` f1; : : : ; ng the sum f(S) = a 1 + \Delta \Delta \Delta + a t (modulo m). We ask: for which...
Accelerated series for universal constants, by the WZ method (2007)
this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and...
And Permutations With, Ira Gessel, Jonathan Weinstein, Herbert S. Wilf
We identify a set of d! signed points, called Toeplitz points, in Z d , with the following property: for every n > 0, the excess of the number of lattice walks of n steps, from the origin to all...
When Are Subset Sums Equidistributed Modulo M? (2007)
Stan Wagon Macalester, Herbert S. Wilf
. For a triple (n; t; m) of positive integers, we attach to each t-subset S = fa 1 ; : : : ; a t g ` f1; : : : ; ng the sum f(S) = a 1 + \Delta \Delta \Delta + a t (modulo m). We ask: for which...
When Are Subset Sums Equidistributed Modulo M? (2007)
Stan Wagon Macalester, Herbert S. Wilf
. For a triple (n, t, m) of positive integers, we attach to each t-subset S = {a 1 ,...,a t }#{1,...,n} the sum f(S)=a 1 + + a t (modulo m). We ask: for which triples (n, t, m)arethe # n t # va l ues...
Jennifer M. Nolan, Herbert S. Wilf, Carla D. Savage
We study basis partitions, introduced by Hansraj Gupta in 1978. For this family of partitions, we give a recurrence, a generating function, identities relating basis partitions to more familiar...
Gert Almkvist, Herbert S. Wilf
The number of partitions of n is given by the asymptotic and convergent series p(n) = 1
Richard Garfield, Herbert S. Wilf
If p is a prime, a is a primitive root modulo p, and n is a positive integer, let r i (n) be the number of k such that 0 k n and \Gamma n
We find all nonnegative integer r, s, p for which the sum � sn k=rn closed form. �pn � k has
We find an interesting relationship between two difficult counting problems in the theory of symmetric functions. For integer k 1, let y k (n) denote the number of standard Young tableaux on n cells...
Let fm (n) denote the number of ways that mn nonattacking kings can be placed on a 2m \Theta 2n chessboard. The purpose of this paper is to prove the following result. Theorem. For each m = 1; 2; 3;...
CONTENTS Chapter 0: What This Book Is About (2007)
copies may be made for classes, etc. Charges, if any, for reproduced copies must be just enough to recover reasonable costs of reproduction. Reproduction for commercial purposes is prohibited. This...
A Discontinuity in the Distribution of Fixed Point Sums (2007)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
The quantity f(n, r), defined as the number of permutations of the set [n] = 2, . . . n} whose fixed points sum to r, shows a sharp discontinuity in the neighborhood of r = n. We explain this...
A Discontinuity in the Distribution of Fixed Point Sums (2007)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
the electronic journal of combinatorics 10 (2003), #R15 1 The quantity f(n, r), defined as the number of permutations of the set [n] = {1, 2,...n} whose fixed points sum to r, shows a sharp...
A Discontinuity in the Distribution of Fixed Point Sums (2007)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
The quantity f(n; r), de ned as the number of permutations of the set [n] = f1; 2; : : : ng whose xed points sum to r, shows a sharp discontinuity in the neighborhood of r = n. We explain this...
Regularly Spaced Subsums of Integer Partitions (2007)
E. Rodney Canfield, Can Eld, Carla D. Savage, Herbert S. Wilf
For integer partitions : n = a 1 + ::: + a k , where a 1 a 2 : : : a k 1, we study the sum a 1 + a 3 + : : : of the parts of odd index. We show that the average of this sum, over all partitions of n,...
Ji, Kathy Q., Wilf, Herbert S.
A recursively palindromic (RP) word is one that is a palindrome and whose left half-word and right half-word are each RP. Thus ABACABA is, and MADAM is not, an RP word. We count RP words of given...
Ewens, Warren J., Wilf, Herbert S.
We present a rapid method for the exact calculation of the cumulative distribution function of the maximum of multinomially distributed random variables. The method runs in time $O(mn)$, where $m$ is...
Counting permutations by their runs up and down (2006)
Canfield, E. Rodney, Wilf, Herbert S.
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
Chen, William Y. C., Ji, Kathy Q., Wilf, Herbert S.
We find the number of partitions of $n$ whose BG-rank is $j$, in terms of $pp(n)$, the number of pairs of partitions whose total number of cells is $n$, giving both bijective and generating function...
We extend classical theorems of Rényi by finding the distributions of the numbers of both weak and strong left-to-right maxima (a.k.a. outstanding elements) in words over a given alphabet and in...
The variance of the Stirling cycle numbers (2005)
We show that the probability that two permutations of $n$ letters have the same number of cycles is \[\sim \frac{1}{2\sqrt{\pi\log{n}}}.\]
Pattern avoidance in compositions and multiset permutations (2005)
Savage, Carla D., Wilf, Herbert S.
We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the...
Mathematical Aspects of Electrical Network Analysis, (2005)
The volume constitutes the final scientific report for a symposium on Mathematical Aspects of Electrical Network Theory. It contains thirteen papers presented during the April 1969 meeting of the...
We show that the probability that two permutations of n letters have the same number of cycles is 1 2 √ π log n. Our purpose here is to prove the following Theorem 1 Let two permutations of n...
The Redheffer matrix of a partially ordered set (2004)
R. Redheffer described an $n\times n$ matrix of 0's and 1's the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and...
The method of characteristics, and "problem 89" of Graham, Knuth and Patashnik (2004)
We apply the method of characteristics for the solution of pde's to two combinatorial problems. The first is finding an explicit form for a distribution that arises in bio-informatics. The second is...
Closed form summation of C-finite sequences (2004)
Greene, Curtis, Wilf, Herbert S.
We consider sums of the form \[\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),\] in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)
Irreducible compositions and the first return to the origin of a random walk (2004)
Bender, Edward A., Lawler, Gregory F., Pemantle, Robin, Wilf, Herbert S.
Let $n = b_1 + ... + b_k = b_1' + \cdot + b_k'$ be a pair of compositions of $n$ into $k$ positive parts. We say this pair is {\em irreducible} if there is no positive $j < k$ for which $b_1 + ......
The combinatorics of a three-line circulant determinant (2003)
Loehr, Nicholas A., Warrington, Gregory S., Wilf, Herbert S.
We study the determinant of the pxp circulant matrix whose first row is (1,-x,0,...,0,-y,0,...,0), the -y being in position q+1. The coefficients of this polynomial are integers that count certain...
Acyclic Digraphs and Eigenvalues of (0,1)-Matrices (2003)
McKay, Brendan D., Oggier, Frederique E., Royle, Gordon F., Sloane, N. J. A., Wanless, Ian M., Wilf, Herbert S.
We show that the number of acyclic directed graphs with n labeled vertices is equal to the number of n X n (0,1)-matrices whose eigenvalues are positive real numbers.
Regularly spaced subsums of integer partitions (2003)
Canfield, E. Rodney, Savage, Carla D., Wilf, Herbert S.
For integer partitions $\lambda :n=a_1+...+a_k$, where $a_1\ge a_2\ge >...\ge a_k\ge 1$, we study the sum $a_1+a_3+...$ of the parts of odd index. We show that the average of this sum, over all...
A Discontinuity in the Distribution of Fixed Point Sums (2003)
Bender, Edward A., Canfield, E. Rodney, Richmond, L. Bruce, Wilf, Herbert S.
The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this...
Some new aspects of the coupon-collector's problem (2003)
Myers, Amy N., Wilf, Herbert S.
We extend the classical coupon collector's problem to one in which two collectors are simultaneously and independently seeking collections of $d$ coupons. We find, in finite terms, the probability...
Longest increasing subsequences in pattern-restricted permutations (2003)
Deutsch, Emeric, Hildebrand, A. J., Wilf, Herbert S.
Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions...
Longest Increasing Subsequences In Pattern-Restricted Permutations (2003)
Emeric Deutsch, A. J. Hildebrand, Herbert S. Wilf
Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we nd those limiting distributions for...
Longest Increasing Subsequences in Pattern-Restricted Permutations (2003)
Emeric Deutsch, A. J. Hildebrand, Herbert S. Wilf
Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions...
The patterns of permutations (2002)
To Dan Kleitman, on his birthday, with all good wishes. Let n, k be positive integers, with k ≤ n, and let τ be a fixed permutation of {1,...,k}. 1 We will call τ the pattern. We will look for...
Searching the web with eigenvectors (2001)
How might we measure the importance of a web site? Well, it’s important if other important web sites link to it (if that sounds circular, just hold on for a moment). Suppose x1,x2,...,xn are the...
The distributions of the entries of Young tableaux (2000)
McKay, Brendan D., Morse, Jennifer, Wilf, Herbert S.
Let T^* be a standard Young tableau of k cells. We show that the probability that a Young tableau of n cells contains T^* as a subtableau is, in the limit n -> \infty, equal to \nu(\pi(T^*))/k!,...
Identically Distributed Pairs of Partition Statistics (2000)
We show that many theorems which assert that two kinds of partitions of the same integer $n$ are equinumerous are actually special cases of a much stronger form of equality. We show that in fact...
The Distributions of the Entries of Young Tableaux 1 (2000)
Brendan D. Mckay, Jennifer Morse, Herbert S. Wilf
Let T be a standard Young tableau of shape l * k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n Q., equal to f l /k!,...
Identically distributed pairs of partition statistics (2000)
We show that many theorems which assert that two kinds of partitions of the same integer n are equinumerous are actually special cases of a much stronger form of equality. We show that in fact there...
series for universal constants, by the WZ method (1999)
In this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and...
Permutation Patterns and Continued Fractions (1999)
Aaron Robertson, Herbert S. Wilf, Doron Zeilberger
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
On the Multiplicity of Parts in a Random Partition (1999)
Sylvie Corteel Laboratoire, Boris Pittel, Carla D. Savage, Herbert S. Wilf
Let be a partition of an integer n chosen uniformly at random among all such partitions. Let s() be a part size chosen uniformly at random from the set of all part sizes that occur in . We prove...
Permutation Patterns and Continued Fractions (1999)
Aaron Robertson, Herbert S. Wilf, Doron Zeilberger
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
Combinatorial Families That Are Exponentially Far From Being Listable In Gray Code Sequence (1999)
Ted Chinburg, Carla D. Savage, Herbert S. Wilf
. Let S(n) be a collection of subsets of f1; :::; ng. In this paper we study numerical obstructions to the existence of orderings of S(n) for which the cardinalities of successive subsets satisfy...
East Side, West Side . . . - an introduction to combinatorial families-with Maple programming (1999)
Contents 1 Introduction 4 1.1 Whatthisisabout ............................ 4 2 About programming in Maple 5 2.1 Exercises.................................. 8 3 Sets and subsets 9 3.1...
Neil Calkin, Herbert S. Wilf, U Q Qqqqqqqq
It is well known (indeed, as Paul Erdős might have said, every child knows) that the rationals are countable. However, the standard presentations of this fact do not give an explicit enumeration;...
Accelerated series for universal constants, by the W Zmethod (1999)
In this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and Zeilberger's...
A combinatorial determinant (1998)
A theorem of Mina evaluates the determinant of a matrix with entries $D^j(f(x)^i)$. We note the important special case where the matrix entries are evaluated at $x=0$ and give a simple proof of it,...
A combinatorial determinant (1998)
A theorem of Mina evaluates the determinant of a matrix with entries D j (f(x) i). We note the important special case where the matrix entries are evaluated at x = 0 and give a simple proof of it, as...
A Pentagonal Number Sieve (1998)
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger
We prove a general "pentagonal sieve" theorem that has corollaries such as the following. First, the number of pairs of partitions of n that have no parts in common is p(n) 2 \Gamma p(n...
On the Multiplicity of Parts in a Random Partition (1998)
Sylvie Corteel, Boris Pittel, Carla D. Savage, Herbert S. Wilf
Let be a partition of an integer n chosen uniformly at random among all such partitions. Let s() be a part size chosen uniformly at random from the set of all part sizes that occur in . We prove...
Carla D. Savage, Boris Pittel, Herbert S. Wilf
Let λ be a partition of an integer n chosen uniformly at random among all such partitions. Let s(λ) be a part size chosen uniformly at random from the set of all part sizes that occur in λ. We...
Sylvie Corteel, Boris Pittel Y, Herbert S. Wilf
Let be a partition of an integer n chosen uniformly at random among all such partitions. Let s ( ) be a part size chosen uniformly at random from the set of all part sizes that occur in. We prove...
Donald E. Knuth, Doron Zeilberger, Herbert S. Wilf
[50] Develop computer programs for simplifying sums
How to do MONTHLY problems with your computer (1997)
Istvan Nemes, Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger
this article (PWZ) have just written a book [8] that describes the theoretical foundations of the solution of this problem, and also gives the software by means of which everyone can perform these...
Lattice walks in Z^d and permutations with no long ascending subsequences (1997)
Ira Gessel, Jonathan Weinstein, Herbert S. Wilf
We identify a set of d! signed points, called Toeplitz points,inZ d , with the following property: for every n>0, the excess of the number of lattice walks of n steps, from the origin to all...
How to do MONTHLY problems with your computer (1997)
István Nemes, Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger
this article (PWZ) have just written a book [8] that describes the theoretical foundations of the solution of this problem, and also gives the software by means of which everyone can perform these...
Donald E. Knuth, Doron Zeilberger, Herbert S. Wilf
This page intentionally left blank [50] Develop computer programs for simplifying sums
Donald E. Knuth, Doron Zeilberger, Herbert S. Wilf
This page intentionally left blank [50] Develop computer programs for simplifying sums
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, Communicated George Andrews
We prove a general ``pentagonal sieve' ' theorem that has corollaries such as the following. First, the number of pairs of partitions of n that have no parts in common is p(n) 2 &...
A High-Tech Proof of the Mills-Robbins-Rumsey Determinant Formula (1995)
Marko Petkovsek, Herbert S. Wilf
this paper. Hence any reader who wishes to do so can cut-and-paste it from there and verify that it does indeed certify the recurrence. It can also be computed by using the Paule-Schorn...
A High-Tech Proof of the Mills-Robbins-Rumsey Determinant Formula (1995)
Marko Petkovsek, Herbert S. Wilf
this paper. Hence any reader who wishes to do so can cut-and-paste it from there and verify that it does indeed certify the recurrence. It can also be computed by using the Paule-Schorn...
On the Outstanding Elements of Permutations (1995)
ach of these j's. Then the probability that a permutation of n letters does have an outstanding element at each of the j # that is marked `Y', and does not have an outstanding element at...
Counting pairs of lattice paths by intersections (1994)
Gessel, Ira, Goddard, Wayne, Shur, Walter, Wilf, Herbert S., Yen, Lily
On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the...
Algorithms and Complexity (1994)
Introduction The network flow problem is an example of a beautiful theoretical subject that has many important applications. It also has generated algorithmic questions that have been in a state of...
Algorithms and Complexity (1994)
Introduction In the previous chapter we met two computational problems for which fast algorithms have never been found, but neither have such algorithms been proved to be unattainable. Those were the...
Algorithms and Complexity (1994)
s, May 1980, 387-406. A more complete account, together with the complexity analysis, is in L. M. Adleman, C. Pomerance and R. S. Rumely, On distinguishing prime numbers from composite numbers,...
Algorithms and Complexity (1994)
y as possible. But that's not the whole story either. There are close connections between algorithmic problems in the theory of numbers, and problems in other fields, seemingly far removed from...
Rational function certification of multisum/integral/``$q$'' identities (1992)
Wilf, Herbert S., Zeilberger, Doron
The method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and ``$q$'' integral/sums.
Rational function certification of multisum /integral/"q" identites (1992)
Herbert S. Wilf, Doron Zeilberger
Abstract. The method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and “q ” integral/sums....
Herbert S. Wilf, Doron Zeilberger
this paper we show that these fast algorithms can be extended to the much larger class of multisum terminating hypergeometric (or equivalently, binomial coefficient) identities, to constant term...
Sums of Closed Form Functions Satisfy Recurrence Relations (1991)
We give a short, self-contained proof of the fact that sums of closed form (hypergeometric) functions satisfy recurrence relations with polynomial coefficients. We obtain explicit bounds for the...
Generatingfunctionology (1990)
This Internet Edition may be reproduced for any valid educational purpose of an institution of higher learning, in which case only the reasonable costs of reproduction may be charged. Reproduction...
Rational functions certify combinatorial identities (1990)
Herbert S. Wilf, Doron Zeilberger
This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present acerti cate that consists of a pair of functions of two integer...
Rational Functions Certify Combinatorial Identities (1989)
Herbert S. Wilf, Doron Zeilberger
This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present a certificate that consists of a pair of functions of two integer...
Algorithms and Complexity (1986)
ly as possible. But that's not the whole story either. There are close connections between algorithmic problems in the theory of numbers, and problems in other fields, seemingly far removed from...
Algorithms and Complexity (1986)
Introduction Chapter 2: Recursive Algorithms 2.1 Introduction Here are two di#erent ways to define n!, if n is a positive integer. The first definition is nonrecursive, the second is recursive. (1)...
Algorithms and Complexity (1986)
Introduction In the previous chapter we met two computational problems for which fast algorithms have never been found, but neither have such algorithms been proved to be unattainable. Those were the...
Algorithms and Complexity (1986)
Introduction Chapter 5: NP-completeness 5.1 Introduction In the previous chapter we met two computational problems for which fast algorithms have never been found, but neither have such algorithms...
Combinatorial algorithms / Albert Nijenhuis and Herbert S. Wilf (1975)
Nijenhuis, Albert, Wilf, Herbert S
Incluye bibliografía e índice
Finite sections of some classical inequalities / [by] Herbert S. Wilf (1970)
Incluye bibliografía
A problem in spectral theory [microform] / (1953)
Thesis (M.A.)--Columbia University, 1953.
Ewens, Warren J., Wilf, Herbert S.
We present a rapid method for the exact calculation of the cumulative distribution function of the maximum of multinomially distributed random variables. The method runs in time O(mn), where m is the...
Jennifer M. Nolan, Carla D. Savage, Herbert S. Wilf
We study basis partitions, introduced by Hansraj Gupta in 1978. For this family of partitions, we give a recurrence, a generating function, identities relating basis partitions to more familiar...
A theorem of Mina evaluates the determinant of a matrix with entries D j (f(x) i ). We note the important special case where the matrix entries are evaluated at x = 0 and give a simple proof of it,...
The distributions of the entries of Young tableaux
Brendan Mckay Jennifer, Jennifer Morse, Herbert S. Wilf
Let T be a standard Young tableau of shape # k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n ##, equal to f /k!, where f...