Abstract
A partition \(\alpha \) is said to contain another partition (or pattern) \(\mu \) if the Ferrers board for \(\mu \) is attainable from \(\alpha \) under removal of rows and columns. We say \(\alpha \) avoids \(\mu \) if it does not contain \(\mu \). In this paper we count the number of partitions of n avoiding a fixed pattern \(\mu \), in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever \(\mu \) is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic p-groups. We further obtain asymptotics for the number of partitions of n avoiding a pattern \(\mu \). Using these asymptotics we conclude that the generating function for \(\mu \) is not algebraic whenever \(\mu \) is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1.
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References
Andrews, G.: Stacked lattice boxes. Ann. Comb. 3(2–4), 115–130 (1999)
Bloom, J., Saracino, D.: Rook and Wilf equivalence of integer partitions. Eur. J. Comb. 71, 246–267 (2018)
Bloom, J., Saracino, D.: On criteria for rook equivalence of Ferrers boards. Eur. J. Comb. 76, 199–207 (2019)
Estermann, T.: On the representations of a number as the sum of two products. J. Lond. Math. Soc. 5(2), 131–137 (1930)
Estermann, T.: On the representations of a number as the sum of three or more products. Proc. Lond. Math. Soc. 34(3), 190–195 (1932)
Fatou, P.: Séries trigonométriques et séries de Taylor. Acta Math. 30(1), 335–400 (1906)
Foata, D., Schützenberger, M.P.: On the rook polynomials of Ferrers relations. In: Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), pp. 413–436. North-Holland, Amsterdam (1970)
Goldman, J., Joichi, J., White, D.: Rook theory. I. Rook equivalence of Ferrers boards. Proc. Am. Math. Soc. 52, 485–492 (1975)
Halberstam, H.: An asymptotic formula in the theory of numbers. Trans. Am. Math. Soc. 84, 338–351 (1957)
Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford University Press, Oxford, sixth edition: Revised by Heath-Brown, D.R., Silverman, J.H. With a foreword by Andrew Wiles (2008)
Huxley, M.N.: Exponential sums and lattice points. III. Proc. Lond. Math. Soc. 87(3), 591–609 (2003)
Ingham, A.E.: Some asymptotic formulae in the theory of numbers. J. Lond. Math. Soc. 2(3), 202–208 (1927)
Johnson, S.M.: On the representations of an integer as the sum of products of integers. Trans. Am. Math. Soc. 76, 177–189 (1954)
Kaplansky, I., Riordan, J.: The problem of the rooks and its applications. Duke Math. J. 13, 259–268 (1946)
Liedahl, S.: Enumeration of metacyclic \(p\)-groups. J. Algebra 186(2), 436–446 (1996)
MacMahon, P.A.: Divisors of numbers and their continuations in the theory of partitions. Proc. Lond. Math. Soc. 19(1), 75–113 (1920)
Nathanson, M.: Partitions with parts in a finite set. Proc. Am. Math. Soc. 128(5), 1269–1273 (2000)
Ramírez Alfonsín, J.L.:. The Diophantine Frobenius problem, vol. 30 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2005)
Stanley, R.: Enumerative combinatorics. Volume 1, Second, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge (2012)
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Bloom, J., McNew, N. Counting pattern-avoiding integer partitions. Ramanujan J 55, 555–591 (2021). https://doi.org/10.1007/s11139-020-00287-6
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DOI: https://doi.org/10.1007/s11139-020-00287-6