Rook theory, compositions, and zeta functions

Haglund, James

doi:10.1090/S0002-9947-96-01662-5

Rook theory, compositions, and zeta functions
HTML articles powered by AMS MathViewer

by James Haglund PDF

Trans. Amer. Math. Soc. 348 (1996), 3799-3825 Request permission

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 $\mathrm {Re}(s)>1/2$. Some identities in the ring of formal power series involving rook theory and continued fractions are developed.

References

Krishnaswami Alladi, Some new observations on the Göllnitz-Gordon and Rogers-Ramanujan identities, Trans. Amer. Math. Soc. 347 (1995), no. 3, 897–914. MR 1284910, DOI 10.1090/S0002-9947-1995-1284910-4
George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
George E. Andrews, The theory of compositions. I. The ordered factorizations of $n$ and a conjecture of C. Long, Canad. Math. Bull. 18 (1975), no. 4, 479–484. MR 419341, DOI 10.4153/CMB-1975-087-0
George E. Andrews, The theory of compositions. II. Simon Newcomb’s problem, Utilitas Math. 7 (1975), 33–54. MR 419342
George E. Andrews, The theory of compositions. II. Simon Newcomb’s problem, Utilitas Math. 7 (1975), 33–54. MR 419342
L. Carlitz, Extended Bernoulli and Eulerian numbers, Duke Math. J. 31 (1964), 667–689. MR 168534, DOI 10.1215/S0012-7094-64-03165-5
H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23-46.
J. F. Dillon and D. P. Roselle, Simon Newcomb’s problem, SIAM J. Appl. Math. 17 (1969), 1086–1093. MR 256898, DOI 10.1137/0117099
D. Foata and M. P. Schützenberger, On the rook polynomials of Ferrers relations, Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969) North-Holland, Amsterdam, 1970, pp. 413–436. MR 0360288
A. Granville, Harald Cramér and the distribution of prime numbers, Scand. Actuar. J. 1995, 12–28.
Jay R. Goldman, J. T. Joichi, and Dennis E. White, Rook theory. I. Rook equivalence of Ferrers boards, Proc. Amer. Math. Soc. 52 (1975), 485–492. MR 429578, DOI 10.1090/S0002-9939-1975-0429578-4
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
J. Haglund, Rook Placements, Compositions, and Permutations of Vectors, Doctoral Dissertation, University of Georgia, Athens, Georgia, 1993.
J. Haglund, Compositions and rook placements, preprint, 1994.
J. Haglund, Compositions and $q$-rook polynomials, in preparation.
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
P. A. MacMahon, Combinatory Analysis, Vol. 1, Cambridge University Press, 1915.
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Don Rawlings, The $(q,\,r)$-Simon Newcomb problem, Linear and Multilinear Algebra 10 (1981), no. 3, 253–260. MR 630153, DOI 10.1080/03081088108817417
John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096594
Richard K. Guy (editor), Reviews in Number Theory 1973-83, Vol. 1a, Amer. Math. Soc., Providence, RI, 1984.
Junya Satoh, $q$-analogue of Riemann’s $\zeta$-function and $q$-Euler numbers, J. Number Theory 31 (1989), no. 3, 346–362. MR 993908, DOI 10.1016/0022-314X(89)90078-4
R. Simion, On Compositions of Multisets, Doctoral Dissertation, The University of Pennsylvania, 1981.
Rodica Simion, A multi-indexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences, J. Combin. Theory Ser. A 36 (1984), no. 1, 15–22. MR 728500, DOI 10.1016/0097-3165(84)90075-X
Richard P. Stanley, Enumerative combinatorics. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo Rota. MR 847717, DOI 10.1007/978-1-4615-9763-6
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550