Enumeration of alternating sign triangles using a constant term approach

Fischer, Ilse

doi:10.1090/tran/7652

Enumeration of alternating sign triangles using a constant term approach
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Trans. Amer. Math. Soc. 372 (2019), 1485-1508 Request permission

Abstract:

Alternating sign triangles (ASTs) have recently been introduced by Ayyer, Behrend, and the author, and it was proven that there is the same number of ASTs with $n$ rows as there is of $n \times n$ alternating sign matrices (ASMs). We prove a conjecture by Behrend on a refined enumeration of ASTs with respect to a statistic that is shown to have the same distribution as the column of the unique $1$ in the top row of an ASM. The proof of the conjecture is based on a certain multivariate generating function of ASTs that takes the positions of the columns with sum $1$ ($1$-columns) into account. We also prove a curious identity on the cyclic rotation of the $1$-columns of ASTs. Furthermore, we discuss a relation of our multivariate generating function to a formula of Di Francesco and Zinn-Justin for the number of fully packed loop configurations associated with a given link pattern. The proofs of our results employ the author’s operator formula for the number of monotone triangles with prescribed bottom row. This is in contrast with the six-vertex model approach that was used by Ayyer, Behrend, and the author to enumerate ASTs, and since the refined enumeration implies the unrefined enumeration, the present paper also provides an alternative proof of the enumeration of ASTs.

References

A. Ayyer, R. E. Behrend, and I. Fischer, Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order, preprint, arXiv:1611.03823, 2016.
Florian Aigner, Refined enumerations of alternating sign triangles, Sém. Lothar. Combin. 78B (2017), Art. 60, 12. MR 3678642
Philippe Biane and Hayat Cheballah, Gog, Magog and Schützenberger II: left trapezoids, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013, pp. 349–360 (English, with English and French summaries). MR 3090996
Roger E. Behrend, Philippe Di Francesco, and Paul Zinn-Justin, On the weighted enumeration of alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 119 (2012), no. 2, 331–363. MR 2860598, DOI 10.1016/j.jcta.2011.09.004
Roger E. Behrend, Philippe Di Francesco, and Paul Zinn-Justin, A doubly-refined enumeration of alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 120 (2013), no. 2, 409–432. MR 2995049, DOI 10.1016/j.jcta.2012.09.004
Jérémie Bettinelli, A simple explicit bijection between $(n,2)$-Gog and Magog trapezoids, Sém. Lothar. Combin. 75 ([2015–2019]), Art. B75e, 9. MR 3504415
R. E. Behrend and I. Fischer, Alternating sign trapezoids and cyclically symmetric lozenge tilings of hexagons with a central triangular hole. In preparation.
David M. Bressoud, Proofs and confirmations, MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999. The story of the alternating sign matrix conjecture. MR 1718370
Hayat Cheballah and Philippe Biane, Gog and Magog triangles, and the Schützenberger involution, Sém. Lothar. Combin. 66 (2011/12), Art. B66d, 20. MR 2971013
Luigi Cantini and Andrea Sportiello, Proof of the Razumov-Stroganov conjecture, J. Combin. Theory Ser. A 118 (2011), no. 5, 1549–1574. MR 2771600, DOI 10.1016/j.jcta.2011.01.007
P. Di Francesco and P. Zinn-Justin, Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule, Electron. J. Combin. 12 (2005), Research Paper 6, 27. MR 2134169
Ilse Fischer, The number of monotone triangles with prescribed bottom row, Adv. in Appl. Math. 37 (2006), no. 2, 249–267. MR 2251438, DOI 10.1016/j.aam.2005.03.009
Ilse Fischer, A new proof of the refined alternating sign matrix theorem, J. Combin. Theory Ser. A 114 (2007), no. 2, 253–264. MR 2293090, DOI 10.1016/j.jcta.2006.04.004
Ilse Fischer, The operator formula for monotone triangles—simplified proof and three generalizations, J. Combin. Theory Ser. A 117 (2010), no. 8, 1143–1157. MR 2677680, DOI 10.1016/j.jcta.2010.03.019
Ilse Fischer, Refined enumerations of alternating sign matrices: monotone $(d,m)$-trapezoids with prescribed top and bottom row, J. Algebraic Combin. 33 (2011), no. 2, 239–257. MR 2765324, DOI 10.1007/s10801-010-0243-7
Ilse Fischer, Short proof of the ASM theorem avoiding the six-vertex model, J. Combin. Theory Ser. A 144 (2016), 139–156. MR 3534066, DOI 10.1016/j.jcta.2016.06.007
Tiago Fonseca and Paul Zinn-Justin, On the doubly refined enumeration of alternating sign matrices and totally symmetric self-complementary plane partitions, Electron. J. Combin. 15 (2008), no. 1, Research Paper 81, 35. MR 2411458
Ira Gessel and Gérard Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), no. 3, 300–321. MR 815360, DOI 10.1016/0001-8708(85)90121-5
C. Krattenthaler, A Gog-Magog Conjecture, http://www.mat.univie.ac.at/\~kratt/artikel/magog.html, 1996.
Greg Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. (2) 156 (2002), no. 3, 835–866. MR 1954236, DOI 10.2307/3597283
Bernt Lindström, On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973), 85–90. MR 335313, DOI 10.1112/blms/5.1.85
W. H. Mills, David P. Robbins, and Howard Rumsey Jr., Proof of the Macdonald conjecture, Invent. Math. 66 (1982), no. 1, 73–87. MR 652647, DOI 10.1007/BF01404757
W. H. Mills, David P. Robbins, and Howard Rumsey Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), no. 3, 340–359. MR 700040, DOI 10.1016/0097-3165(83)90068-7
W. H. Mills, David P. Robbins, and Howard Rumsey Jr., Self-complementary totally symmetric plane partitions, J. Combin. Theory Ser. A 42 (1986), no. 2, 277–292. MR 847558, DOI 10.1016/0097-3165(86)90098-1
James Propp, The many faces of alternating-sign matrices, Discrete models: combinatorics, computation, and geometry (Paris, 2001) Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrèt. (MIMD), Paris, 2001, pp. 043–058. MR 1888762
Dan Romik, Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities, Comm. Math. Phys. 330 (2014), no. 2, 499–538. MR 3223480, DOI 10.1007/s00220-014-2001-5
David P. Robbins and Howard Rumsey Jr., Determinants and alternating sign matrices, Adv. in Math. 62 (1986), no. 2, 169–184. MR 865837, DOI 10.1016/0001-8708(86)90099-X
Benjamin Wieland, A large dihedral symmetry of the set of alternating sign matrices, Electron. J. Combin. 7 (2000), Research Paper 37, 13. MR 1773294
Doron Zeilberger, Proof of the alternating sign matrix conjecture, Electron. J. Combin. 3 (1996), no. 2, Research Paper 13, approx. 84. The Foata Festschrift. MR 1392498
Doron Zeilberger, Proof of the refined alternating sign matrix conjecture, New York J. Math. 2 (1996), 59–68, electronic. MR 1383799
P. Zinn-Justin, Integrability and combinatorics: selected topics, Exact methods in low-dimensional statistical physics and quantum computing, Oxford Univ. Press, Oxford, 2010, pp. 483–526. MR 2668655
P. Zinn-Zhyusten and P. Di Franchesko, The quantum Knizhnik-Zamolodchikov equation, completely symmetric self-complementary plane partitions, and alternating-sign matrices, Teoret. Mat. Fiz. 154 (2008), no. 3, 387–408 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 154 (2008), no. 3, 331–348. MR 2431554, DOI 10.1007/s11232-008-0031-x