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March  2021, 29(1): 1841-1857. doi: 10.3934/era.2020094

Skew doubled shifted plane partitions: Calculus and asymptotics

Université de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France

* Corresponding author: Huan Xiong

Received  April 2020 Revised  July 2020 Published  September 2020

Fund Project: The second author was supported by SNSF grant P2ZHP2_171879

In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions "shifted at the two sides". We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using similar methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.

Citation: Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094
References:
[1]

G. E. Andrews, MacMahon's conjecture on symmetric plane partitions, Proc. Nat. Acad. Sci. U.S.A., 74 (1977), 426-429.  doi: 10.1073/pnas.74.2.426.  Google Scholar

[2]

G. E. Andrews, Plane partitions. I. The MacMahon conjecture, in Studies in Foundations and Combinatorics, Adv. in Math. Suppl. Stud., 1, Academic Press, New York-London, 1978,131-150.  Google Scholar

[3]

D. Betea, J. Bouttier, P. Nejjar and M. Vuletić, The free boundary Schur process and applications I, Ann. Henri Poincaré, 19 (2018), 3663–3742. doi: 10.1007/s00023-018-0723-1.  Google Scholar

[4]

A. Borodin, Periodic Schur process and cylindric partitions, Duke Math. J., 140 (2007), 391-468.  doi: 10.1215/S0012-7094-07-14031-6.  Google Scholar

[5]

A. Borodin and I. Corwin, Macdonald processes, Probab. Theory Related Fields, 158 (2014), 225-400.  doi: 10.1007/s00440-013-0482-3.  Google Scholar

[6]

M. Ciucu and C. Krattenthaler, Enumeration of Lozenge tilings of hexagons with cut-off corners, J. Combin. Theory Ser. A, 100 (2002), 201-231.  doi: 10.1006/jcta.2002.3288.  Google Scholar

[7]

S. CorteelC. Savelief and M. Vuletić, Plane overpartitions and cylindric partitions, J. Combin. Theory Ser. A, 118 (2011), 1239-1269.  doi: 10.1016/j.jcta.2010.12.001.  Google Scholar

[8]

M. Dewar and M. R. Murty, An asymptotic formula for the coefficients of $j(z)$, Int. J. Number Theory, 9 (2013), 641-652.  doi: 10.1142/S1793042112501539.  Google Scholar

[9]

W. Fang, H.-K. Hwang and M. Kang, Phase transitions from $\exp(n^{1/2})$ to $\exp(n^{2/3})$ in the asymptotics of banded plane partitions, preprint, arXiv: 2004.08901. Google Scholar

[10]

I. M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amer. Math. Soc., 349 (1997), 429-479.  doi: 10.1090/S0002-9947-97-01791-1.  Google Scholar

[11]

G.-N. Han and H. Xiong, Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions, Adv. in Appl. Math., 96 (2018), 18-38.  doi: 10.1016/j.aam.2017.12.007.  Google Scholar

[12]

A. IqbalS. NazirZ. Raza and Z. Saleem, Generalizations of Nekrasov-Okounkov identity, Ann. Comb., 16 (2012), 745-753.  doi: 10.1007/s00026-012-0157-2.  Google Scholar

[13]

V. Kotěšovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, preprint, arXiv: 1509.08708. Google Scholar

[14]

R. Langer, Enumeration of cylindric plane partitions – Part Ⅱ, preprint, arXiv: 1209.1807. Google Scholar

[15] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.   Google Scholar
[16]

P. A. MacMahon, Partitions of numbers whose graphs possess symmetry, Trans. Cambridge Philos. Soc., 17 (1899), 149-170.   Google Scholar

[17]

N. A. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, in The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006,525–596. doi: 10.1007/0-8176-4467-9_15.  Google Scholar

[18]

A. Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.), 7 (2001), 57-81.  doi: 10.1007/PL00001398.  Google Scholar

[19]

A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc., 16 (2003), 581-603.  doi: 10.1090/S0894-0347-03-00425-9.  Google Scholar

[20]

A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, in The Unity of Mathematics, Springer, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006,597–618. doi: 10.1007/0-8176-4467-9_16.  Google Scholar

[21]

B. E. Sagan, Combinatorial proofs of hook generating functions for skew plane partitions, Theoret. Comput. Sci., 117 (1993), 273-287.  doi: 10.1016/0304-3975(93)90319-O.  Google Scholar

[22]

R. P. Stanley, Theory and application of plane partitions. Ⅰ, Ⅱ, Studies in Appl. Math., 50 (1971), 167–188,259–279. doi: 10.1002/sapm1971502167.  Google Scholar

[23]

R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combinatorial Theory Ser. A, 14 (1973), 53-65.  doi: 10.1016/0097-3165(73)90063-0.  Google Scholar

[24] R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511609589.  Google Scholar
[25]

P. Tingley, Three combinatorial models for $\widehat {\rm sl} _n$ crystals, with applications to cylindric plane partitions, Int. Math. Res. Not. IMRN, 2008 (2008), 40pp. doi: 10.1093/imrn/rnm143.  Google Scholar

[26]

M. Vuletić, The shifted Schur process and asymptotics of large random strict plane partitions, Int. Math. Res. Not. IMRN, 2007 (2007), 53pp. doi: 10.1093/imrn/rnm043.  Google Scholar

[27]

M. Vuletić, A generalization of MacMahon's formula, Trans. Amer. Math. Soc., 361 (2009), 2789-2804.  doi: 10.1090/S0002-9947-08-04753-3.  Google Scholar

show all references

References:
[1]

G. E. Andrews, MacMahon's conjecture on symmetric plane partitions, Proc. Nat. Acad. Sci. U.S.A., 74 (1977), 426-429.  doi: 10.1073/pnas.74.2.426.  Google Scholar

[2]

G. E. Andrews, Plane partitions. I. The MacMahon conjecture, in Studies in Foundations and Combinatorics, Adv. in Math. Suppl. Stud., 1, Academic Press, New York-London, 1978,131-150.  Google Scholar

[3]

D. Betea, J. Bouttier, P. Nejjar and M. Vuletić, The free boundary Schur process and applications I, Ann. Henri Poincaré, 19 (2018), 3663–3742. doi: 10.1007/s00023-018-0723-1.  Google Scholar

[4]

A. Borodin, Periodic Schur process and cylindric partitions, Duke Math. J., 140 (2007), 391-468.  doi: 10.1215/S0012-7094-07-14031-6.  Google Scholar

[5]

A. Borodin and I. Corwin, Macdonald processes, Probab. Theory Related Fields, 158 (2014), 225-400.  doi: 10.1007/s00440-013-0482-3.  Google Scholar

[6]

M. Ciucu and C. Krattenthaler, Enumeration of Lozenge tilings of hexagons with cut-off corners, J. Combin. Theory Ser. A, 100 (2002), 201-231.  doi: 10.1006/jcta.2002.3288.  Google Scholar

[7]

S. CorteelC. Savelief and M. Vuletić, Plane overpartitions and cylindric partitions, J. Combin. Theory Ser. A, 118 (2011), 1239-1269.  doi: 10.1016/j.jcta.2010.12.001.  Google Scholar

[8]

M. Dewar and M. R. Murty, An asymptotic formula for the coefficients of $j(z)$, Int. J. Number Theory, 9 (2013), 641-652.  doi: 10.1142/S1793042112501539.  Google Scholar

[9]

W. Fang, H.-K. Hwang and M. Kang, Phase transitions from $\exp(n^{1/2})$ to $\exp(n^{2/3})$ in the asymptotics of banded plane partitions, preprint, arXiv: 2004.08901. Google Scholar

[10]

I. M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amer. Math. Soc., 349 (1997), 429-479.  doi: 10.1090/S0002-9947-97-01791-1.  Google Scholar

[11]

G.-N. Han and H. Xiong, Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions, Adv. in Appl. Math., 96 (2018), 18-38.  doi: 10.1016/j.aam.2017.12.007.  Google Scholar

[12]

A. IqbalS. NazirZ. Raza and Z. Saleem, Generalizations of Nekrasov-Okounkov identity, Ann. Comb., 16 (2012), 745-753.  doi: 10.1007/s00026-012-0157-2.  Google Scholar

[13]

V. Kotěšovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, preprint, arXiv: 1509.08708. Google Scholar

[14]

R. Langer, Enumeration of cylindric plane partitions – Part Ⅱ, preprint, arXiv: 1209.1807. Google Scholar

[15] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.   Google Scholar
[16]

P. A. MacMahon, Partitions of numbers whose graphs possess symmetry, Trans. Cambridge Philos. Soc., 17 (1899), 149-170.   Google Scholar

[17]

N. A. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, in The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006,525–596. doi: 10.1007/0-8176-4467-9_15.  Google Scholar

[18]

A. Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.), 7 (2001), 57-81.  doi: 10.1007/PL00001398.  Google Scholar

[19]

A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc., 16 (2003), 581-603.  doi: 10.1090/S0894-0347-03-00425-9.  Google Scholar

[20]

A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, in The Unity of Mathematics, Springer, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006,597–618. doi: 10.1007/0-8176-4467-9_16.  Google Scholar

[21]

B. E. Sagan, Combinatorial proofs of hook generating functions for skew plane partitions, Theoret. Comput. Sci., 117 (1993), 273-287.  doi: 10.1016/0304-3975(93)90319-O.  Google Scholar

[22]

R. P. Stanley, Theory and application of plane partitions. Ⅰ, Ⅱ, Studies in Appl. Math., 50 (1971), 167–188,259–279. doi: 10.1002/sapm1971502167.  Google Scholar

[23]

R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combinatorial Theory Ser. A, 14 (1973), 53-65.  doi: 10.1016/0097-3165(73)90063-0.  Google Scholar

[24] R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511609589.  Google Scholar
[25]

P. Tingley, Three combinatorial models for $\widehat {\rm sl} _n$ crystals, with applications to cylindric plane partitions, Int. Math. Res. Not. IMRN, 2008 (2008), 40pp. doi: 10.1093/imrn/rnm143.  Google Scholar

[26]

M. Vuletić, The shifted Schur process and asymptotics of large random strict plane partitions, Int. Math. Res. Not. IMRN, 2007 (2007), 53pp. doi: 10.1093/imrn/rnm043.  Google Scholar

[27]

M. Vuletić, A generalization of MacMahon's formula, Trans. Amer. Math. Soc., 361 (2009), 2789-2804.  doi: 10.1090/S0002-9947-08-04753-3.  Google Scholar

Figure 1.  Various kinds of defective plane partitions
Figure 2.  Asymptotic formulas for various kinds of defective plane partitions
Figure 3.  A skew doubled shifted plane partition
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