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Periods in strings
 Journal of Combinatorial Theory, Series A
, 1981
"... A survey is presented of some methods and results on counting words that satisfy various restrictions on subwords (i.e., blocks of consecutive symbols). Various applications to commafree codes, games, pattern matching, and other subjects are indicated. The emphasis is on the unified treatment of th ..."
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Cited by 78 (0 self)
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A survey is presented of some methods and results on counting words that satisfy various restrictions on subwords (i.e., blocks of consecutive symbols). Various applications to commafree codes, games, pattern matching, and other subjects are indicated. The emphasis is on the unified treatment of those topics through the use of generating functions. 1.
Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
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Cited by 38 (15 self)
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
ASYMPTOTICS OF THE QUANTUM INVARIANTS FOR SURGERIES ON THE FIGURE 8 KNOT
, 2005
"... We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a for ..."
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Cited by 4 (1 self)
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We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2, C)representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern–Simons invariants of the corresponding flat SU(2)connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure 8 knot following Kashaev [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].
On the Number of Distinct Block Sizes in Partitions of a Set
, 1985
"... The average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n ® . In addition, almost all partitions have approximately e log n distinct block sizes. This is in striking contrast to the fact that the average total number of blocks in a partition is ~ ..."
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Cited by 2 (1 self)
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The average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n ® . In addition, almost all partitions have approximately e log n distinct block sizes. This is in striking contrast to the fact that the average total number of blocks in a partition is ~ n(log n)  1 as n ® . On the number of distinct block sizes in partitions of a set A. M. Odlyzko Bell Laboratories Murray Hill, New Jersey 07974 USA and L. B. Richmond Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1 Canada 1. Introduction A recent paper of H. Wilf [6] compares the number of distinct part sizes to the total number of parts in various combinatorial partition problems. It is well known and easy to prove that the average number of cycles of a permutation on n symbols is log n + g + o(1) as n ® , when g = 0.577 ... denotes Euler's constant. Wilf showed that the average number of distinct cycle sizes in a permutation on n let...
Asymptotic behaviour of the poles of a special generating function for acyclic digraphs
, 2005
"... ..."
On the inversion of ... in terms of associated Stirling numbers
, 1995
"... > n+m n m z n n! : (1a) The numbers (\Gamma1) n+m \Theta n m are also called Stirling numbers of the first kind [8]. Stirling partition numbers \Phi n m \Psi , also called Stirling numbers of the second kind, are defined by (e z \Gamma 1) m = m! X n ae n m oe z n n! ; (1 ..."
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> n+m n m z n n! : (1a) The numbers (\Gamma1) n+m \Theta n m are also called Stirling numbers of the first kind [8]. Stirling partition numbers \Phi n m \Psi , also called Stirling numbers of the second kind, are defined by (e z \Gamma 1) m = m! X n ae n m oe z n n! ; (1b) and 2associated Stirling partition numbers \Phi n m \Psi 2 are defined by [2, exercise 5.7; 7, p. 296; 9, x4.5] (e z \Gamma 1 \Gamma z) m = m! X n ae n m
unknown title
, 1995
"... The function y = Φα(x), the solution of y α e y = x for x and y large enough, has a series expansion in terms of lnx and lnlnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x> (αe) α for α ≥ 1. It is also shown that new expansions can be o ..."
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The function y = Φα(x), the solution of y α e y = x for x and y large enough, has a series expansion in terms of lnx and lnlnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x> (αe) α for α ≥ 1. It is also shown that new expansions can be obtained for Φα in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain.