Rook theory, compositions, and zeta functions
Author:
James Haglund
Journal:
Trans. Amer. Math. Soc. 348 (1996), 37993825
MSC (1991):
Primary 11M41, 05A15
MathSciNet review:
1357880
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in . Some identities in the ring of formal power series involving rook theory and continued fractions are developed.
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 K. Alladi, Some New Observations on the GöllnitzGordon and RogersRamanujan Identities, Trans. Amer. Math. Soc. 347 (1995), 897914. MR 95h:11109
 [An1]
 G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, AddisonWesley, Reading, Mass., 1976. MR 58:27738
 [An2]
 G. E. Andrews, The Theory of Compositions (I): The Ordered Factorizations of n and a conjecture of C. Long, Canad. Math. Bull. 18 (1975), 479484. MR 54:7363
 [An3]
 G. E. Andrews, The Theory of Compositions (II): Simon Newcomb's Problem, Utilitas Math. 7 (1974), 3354. MR 54:7364a
 [An4]
 G. E. Andrews, The Theory of Compositions (III): the MacMahon Formula and the StantonCowan Numbers, Utilitas Math. 9 (1976), 283290. MR 54:7364b
 [Car]
 L. Carlitz, Extended Bernoulli and Eulerian numbers, Duke Math. J. 31 (1964), 667689. MR 29:5796
 [Cra]
 H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 2346.
 [DR]
 J. F. Dillon and D. P. Roselle, Simon Newcomb's Problem, SIAM J. Appl. Math. 17 (1969), 10861093. MR 41:1553
 [FS]
 D. C. Foata and M. P. Schützenberger, On the rook polynomials of Ferrers relations, Colloq. Math. Soc. Janos Bolyai, 4, Combinatorial Theory and its Applications, vol. 2, NorthHolland, Amsterdam, 1970, pp. (413436). MR 50:12738
 [Gra]
 A. Granville, Harald Cramér and the distribution of prime numbers, Scand. Actuar. J. 1995, 1228. CMP 95:17
 [GJW]
 J. Goldman, J. Joichi, and D. White, Rook theory I: Rook equivalence of Ferrers boards, Proc. Amer. Math. Soc. 52 (1975), 485492. MR 55:2590
 [HW]
 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 1979. MR 81i:10002
 [Ha1]
 J. Haglund, Rook Placements, Compositions, and Permutations of Vectors, Doctoral Dissertation, University of Georgia, Athens, Georgia, 1993.
 [Ha2]
 J. Haglund, Compositions and rook placements, preprint, 1994.
 [Ha3]
 J. Haglund, Compositions and rook polynomials, in preparation.
 [KR]
 I. Kaplansky and J. Riordan, The problem of the rooks and its applications, Duke Math. J. 13 (1946), 259268. MR 7:508d
 [Mac]
 P. A. MacMahon, Combinatory Analysis, Vol. 1, Cambridge University Press, 1915.
 [Per]
 O. Perron, Die Lehre von den Kettenbrüchen, Chelsea, New York, 1950. MR 12:254b
 [Raw]
 D. Rawlings, The Simon Newcomb Problem, Linear and Multilinear Algebra 10 (1981), 253260. MR 83k:05009
 [Rio]
 J. Riordan, An Introduction to Combinatorial Analysis, John Wiley, New York, 1958. MR 20:3077
 [RNT]
 Richard K. Guy (editor), Reviews in Number Theory 197383, Vol. 1a, Amer. Math. Soc., Providence, RI, 1984.
 [Sat]
 J. Satoh, Analogue of Riemann's function and Euler numbers, J. Number Theory 31 (1989), 346362. MR 90d:11132
 [Si1]
 R. Simion, On Compositions of Multisets, Doctoral Dissertation, The University of Pennsylvania, 1981.
 [Si2]
 R. Simion, A multiindexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences, J. Combin. Theory Ser. A 36 (1984), 1522. MR 85e:05015
 [Sta]
 R. P. Stanley, Enumerative Combinatorics: Volume I, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986. MR 87j:05003
 [Tic]
 E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, Second Edition, Clarendon Press, 1986. MR 88c:11049
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Additional Information
James Haglund
Affiliation:
Department of Mathematics, The University of Illinois at UrbanaChampaign, Urbana, IL 61801
Email:
jhaglund@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S0002994796016625
PII:
S 00029947(96)016625
Keywords:
Riemann zeta function,
Riemann hypothesis,
continued fraction,
composition,
rook theory,
Euler product
Received by editor(s):
January 20, 1995
Received by editor(s) in revised form:
November 6, 1995
Article copyright:
© Copyright 1996
American Mathematical Society
