Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-13T21:26:48.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit Shapes via Bijections

Part of: Combinatorics Combinatorial probability Additive number theory; partitions Limit theorems

Published online by Cambridge University Press:  02 August 2018

STEPHEN DeSALVO  and
IGOR PAK
STEPHEN DeSALVO
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: stephendesalvo@math.ucla.edu, pak@math.ucla.edu)
IGOR PAK
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: stephendesalvo@math.ucla.edu, pak@math.ucla.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 05A19: Combinatorial identities, bijective combinatorics 60C05: Combinatorial probability 11P82: Analytic theory of partitions 11P84: Partition identities; identities of Rogers-Ramanujan type 60F10: Large deviations
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 187 - 240
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alon, N. and Krivelevich, M. (2008) Extremal and probabilistic combinatorics. In Princeton Companion to Mathematics (Gowers, W. T.et al., eds), Princeton University Press, pp. 562–575.Google Scholar
[2] Andrews, G. E. (1984) The Theory of Partitions, Cambridge University Press.Google Scholar
[3] Andrews, G. E. (2000) MacMahon's partition analysis, II: Fundamental theorems. Ann. Comb. 4 327–338.Google Scholar
[4] Arratia, R. and DeSalvo, S. (2016) Probabilistic divide-and-conquer: A new exact simulation method, with integer partitions as an example. Combin. Probab. Comput. 25 324–351.Google Scholar
[5] Arratia, R. and Tavaré, S. (1994) Independent process approximations for random combinatorial structures. Adv. Math. 104 90–154.Google Scholar
[6] Auluck, F. C. and Haselgrove, C. B. (1952) On Ingham's Tauberian theorem for partitions. Proc. Cambridge Philos. Soc. 48 566–570.Google Scholar
[7] Bateman, P. T. and Erdős, P. (1956) Monotonicity of partition functions. Mathematika 3 1–14.Google Scholar
[8] Blackburn, S. R., Neumann, P. M. and Venkataraman, G. (2007) Enumeration of Finite Groups, Cambridge University Press.Google Scholar
[9] Bobrowski, A. (2005) Functional Analysis for Probability and Stochastic Processes, Cambridge University Press.Google Scholar
[10] Bogachev, L. V. (2015) Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts. Random Struct. Alg. 47 227–266.Google Scholar
[11] de Bruijn, N. G. (1948) On Mahler's partition problem. Indag. Math. 10 210–220.Google Scholar
[12] Canfield, E. R., Corteel, S. and Hitczenko, P. (2002) Random partitions with non negative rth differences. In LATIN 2002: Theoretical Informatics (Rajsbaum, S., ed.), Vol. 2286 of Lecture Notes in Computer Science, Springer, pp. 131–140.Google Scholar
[13] Canfield, E. R. and Wilf, H. S. (2012) On the growth of restricted integer partition functions. In Partitions, q-series, and Modular Forms (Alladi, K. and Garvan, F., eds), Springer, pp. 39–46.Google Scholar
[14] Cohn, H., Elkies, N. and Propp, J. (1996) Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85 117–166.Google Scholar
[15] Colomo, F. and Pronko, A. G. (2010) The limit shape of large alternating sign matrices. SIAM J. Discrete Math. 24 1558–1571.Google Scholar
[16] Corteel, S. and Savage, C. D. (2004) Partitions and compositions defined by inequalities. Ramanujan J. 8 357–381.Google Scholar
[17] Corteel, S., Savage, C. D. and Wilf, H. S. (2005) A note on partitions and compositions defined by inequalities. Integers 5 A24.Google Scholar
[18] Dembo, A., Vershik, A. and Zeitouni, O. (2000) Large deviations for integer partitions. Markov Process. Rel. Fields 6 147–179.Google Scholar
[19] Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, Springer.Google Scholar
[20] DeSalvo, S. (2018) Probabilistic divide-and-conquer: Deterministic second half. Adv. Appl. Math. 92 17–50.Google Scholar
[21] Erdős, P. and Lehner, J. (1941) The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8 335–345.Google Scholar
[22] Erdős, P. and Szalay, M. (1984) On the statistical theory of partitions. Colloq. Math. Soc. János Bolyai 34 397–450.Google Scholar
[23] Freiman, G. A., Vershik, A. M. and Yakubovich, Y. V. (2000) A local limit theorem for random strict partitions. Theory Probab. Appl. 44 453–468.Google Scholar
[24] Fristedt, B. (1993) The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337 703–735.Google Scholar
[25] Fulman, J. and Goldstein, L. (2001) Zero biasing and Jack measures. Combin. Probab. Comput. 20 753–762.Google Scholar
[26] Garsia, A. M. and Milne, S. C. (1981) A Rogers–Ramanujan bijection. J. Combin. Theory Ser. A 31 289–339.Google Scholar
[27] Goh, W. M. Y. and Hitczenko, P. (2008) Random partitions with restricted part sizes. Random Struct. Alg. 32 440–462.Google Scholar
[28] Granovsky, B. L., Stark, D. and Erlihson, M. (2008) Meinardus' theorem on weighted partitions: Extensions and a probabilistic proof. Adv. Appl. Math. 41 307–328.Google Scholar
[29] Hardy, G. H. and Ramanujan, S. (1918) Asymptotic formulas in combinatory analysis. Proc. London Math. Soc. 2 75–115.Google Scholar
[30] Haselgrove, C. B. and Temperley, H. N. V. (1954) Asymptotic formulae in the theory of partitions. Proc. Cambridge Philos. Soc. 5 225–241.Google Scholar
[31] Hwang, H. K. (2001) Limit theorems for the number of summands in integer partitions. J. Combin. Theory Ser. A 96 89–126.Google Scholar
[32] Ingham, A. E. (1941) A Tauberian theorem for partitions. Ann. of Math. 42 1075–1090.Google Scholar
[33] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.Google Scholar
[34] Kenyon, R., Okounkov, A. and Sheffield, S. (2006) Dimers and amoebae. Ann. of Math. 163 1019–1056.Google Scholar
[35] Khinchin, A. I. (1949) Mathematical Foundations of Statistical Mechanics, Dover.Google Scholar
[36] Konvalinka, M. and Pak, I. (2009) Geometry and complexity of O'Hara's algorithm. Adv. Appl. Math. 42 157–175.Google Scholar
[37] Konvalinka, M. and Pak, I. (2014) Cayley compositions, partitions, polytopes, and geometric bijections. J. Combin. Theory Ser. A 123 86–91.Google Scholar
[38] Madritsch, M. and Wagner, S. (2010) A central limit theorem for integer partitions. Monatsh. Math. 161 85–114.Google Scholar
[39] Meinardus, G. (1954) Asymptotische Aussagen über Partitionen. Math. Z. 59 388–398.Google Scholar
[40] von Neumann, J. (1951) Various techniques used in connection with random digits. J. Res. Nat. Bur. Stand. Appl. Math. Ser. 12 36–38.Google Scholar
[41] O'Hara, K. M. (1988) Bijections for partition identities. J. Combin. Theory Ser. A 49 13–25.Google Scholar
[42] Pak, I. (2004) Partition identities and geometric bijections. Proc. Amer. Math. Soc. 132 3457–3462.Google Scholar
[43] Pak, I. (2004) The nature of partition bijections, II: Asymptotic stability. math.ucla.edu/~pak/papers/stab5.pdfGoogle Scholar
[44] Pak, I. (2006) Partition bijections: A survey. Ramanujan J. 12 5–75.Google Scholar
[45] Pittel, B. (1997) On a likely shape of the random Ferrers diagram. Adv. Appl. Math. 18 432–488.Google Scholar
[46] Pittel, B. (1999) Confirming two conjectures about the integer partitions. J. Combin. Theory Ser. A 88 123–135.Google Scholar
[47] Romik, D. (2003) Identities arising from limit shapes of constrained random partitions. math.ucdavis.edu/~romik/data/uploads/papers/shape.pdfGoogle Scholar
[48] Romik, D. (2005) Partitions of n into t $\sqrt {n}$ parts. European J. Combin. 26 1–17.Google Scholar
[49] Romik, D. (2012) Arctic circles, domino tilings and square Young tableaux. Ann. Probab. 40 611–647.Google Scholar
[50] Romik, D. (2015) The Surprising Mathematics of Longest Increasing Subsequences, Cambridge University Press.Google Scholar
[51] Roth, K. F. and Szekeres, G. (1954) Some asymptotic formulae in the theory of partitions. Quart. J. Math. 5 241–259.Google Scholar
[52] Sloane, N. J. A. The Online Encyclopedia of Integer Sequences. oeis.orgGoogle Scholar
[53] Stanton, D. (2009) q-analogues of Euler's odd = distinct theorem. Ramanujan J. 19 107–113.Google Scholar
[54] Temperley, H. N. V. (1952) Statistical mechanics and the partition of numbers, II: The form of crystal surfaces. Proc. Cambridge Philos. Soc. 48 683–697.Google Scholar
[55] Vershik, A. (1996) Statistical mechanics of combinatorial partitions, and their limit shapes. Funct. Anal. Appl. 30 90–105.Google Scholar
[56] Vershik, A. M. and Kerov, S. V. (1977) Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233 1024–1027.Google Scholar
[57] Vershik, A. and Yakubovich, Y. (2001) The limit shape and fluctuations of random partitions of naturals with fixed number of summands. Mosc. Math. J. 1 457–468.Google Scholar
[58] Wright, E. M. (1934) Asymptotic partition formulae, III: Partitions into k-th powers. Acta Math. 63 143–191.Google Scholar
[59] Yakubovich, Y. (2012) Ergodicity of multiplicative statistics. J. Combin. Theory Ser. A 119 1250–1279.Google Scholar
[60] Yee, Ae Ja (2004) Combinatorial proofs of Ramanujan's 1ψ1 summation and the q-Gauss summation. J. Combin. Theory Ser. A 105 63–77.Google Scholar