Browse issues
no. 5
Hook formulas for skew shapes III. Multivariate and product formulas
Morales, Alejandro H.1; Pak, Igor2; Panova, Greta3
1 University of Massachusetts, Amherst Department of Mathematics and Statistics Amherst, MA, 01002, USA
2 University of California, Los Angeles Department of Mathematics Los Angeles, CA, 90095, USA
3 University of Southern California Department of Mathematics Los Angeles, CA, 90089, USA
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 815-861.

We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series [54, 52]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.67
Keywords: skew standard tableaux, product formulas, hook length, lozenge tilings, Schubert polynomials

Morales, Alejandro H. 1; Pak, Igor 2; Panova, Greta 3

1 University of Massachusetts, Amherst Department of Mathematics and Statistics Amherst, MA, 01002, USA
2 University of California, Los Angeles Department of Mathematics Los Angeles, CA, 90095, USA
3 University of Southern California Department of Mathematics Los Angeles, CA, 90089, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2019__2_5_815_0,
     author = {Morales, Alejandro H. and Pak, Igor and Panova, Greta},
     title = {Hook formulas for skew shapes {III.} {Multivariate} and product formulas},
     journal = {Algebraic Combinatorics},
     pages = {815--861},
     publisher = {MathOA foundation},
     volume = {2},
     number = {5},
     year = {2019},
     doi = {10.5802/alco.67},
     mrnumber = {4023568},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.67/}
}
TY  - JOUR
AU  - Morales, Alejandro H.
AU  - Pak, Igor
AU  - Panova, Greta
TI  - Hook formulas for skew shapes III. Multivariate and product formulas
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 815
EP  - 861
VL  - 2
IS  - 5
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.67/
DO  - 10.5802/alco.67
LA  - en
ID  - ALCO_2019__2_5_815_0
ER  - 
%0 Journal Article
%A Morales, Alejandro H.
%A Pak, Igor
%A Panova, Greta
%T Hook formulas for skew shapes III. Multivariate and product formulas
%J Algebraic Combinatorics
%D 2019
%P 815-861
%V 2
%N 5
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.67/
%R 10.5802/alco.67
%G en
%F ALCO_2019__2_5_815_0
Morales, Alejandro H.; Pak, Igor; Panova, Greta. Hook formulas for skew shapes III. Multivariate and product formulas. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 815-861. doi : 10.5802/alco.67. https://alco.centre-mersenne.org/articles/10.5802/alco.67/

[1] Adin, Ron; Roichman, Yuval Standard Young tableaux, Handbook of enumerative combinatorics (Discrete Math. Appl. (Boca Raton)), CRC Press, Boca Raton, FL, 2015, pp. 895-974 | DOI | MR | Zbl

[2] Ardila, Federico; Serrano, Luis G. Staircase skew Schur functions are Schur P-positive, J. Algebraic Combin., Volume 36 (2012) no. 3, pp. 409-423 | DOI | MR | Zbl

[3] Askey, Richard A.; Roy, Ranjan Barnes G function, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010

[4] Austen, Jane Pride and prejudice, I.–III, T. Edgerton, Whitehall, London, England, 1813

[5] Baryshnikov, Yuliy; Romik, Dan Enumeration formulas for Young tableaux in a diagonal strip, Israel J. Math., Volume 178 (2010), pp. 157-186 | DOI | MR | Zbl

[6] Betea, Dan Elliptic Combinatorics and Markov Processes, Ph. D. Thesis, California Institute of Technology (USA) (2012), 124 pages | MR

[7] Billey, Sara C.; Holroyd, Alexander E.; Young, Benjamin J. A bijective proof of Macdonald’s reduced word formula, Algebr. Comb., Volume 2 (2019) no. 2, pp. 217-248 | MR | Zbl

[8] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 345-374 | DOI | MR | Zbl

[9] Borodin, Alexei; Gorin, Vadim Shuffling algorithm for boxed plane partitions, Adv. Math., Volume 220 (2009) no. 6, pp. 1739-1770 | DOI | MR | Zbl

[10] Borodin, Alexei; Gorin, Vadim; Rains, Eric M. q-distributions on boxed plane partitions, Sel. Math. (N.S.), Volume 16 (2010) no. 4, pp. 731-789 | DOI | MR | Zbl

[11] Bressoud, David M. Proofs and confirmations. The story of the alternating sign matrix conjecture, MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999, xvi+274 pages | MR | Zbl

[12] Cohn, Henry; Larsen, Michael; Propp, James The shape of a typical boxed plane partition, New York J. Math., Volume 4 (1998), pp. 137-165 | MR | Zbl

[13] DeWitt, Elizabeth Angela Identities Relating Schur s-Functions and Q-Functions, Ph. D. Thesis, University of Michigan (USA) (2012), 67 pages | MR

[14] Dousse, Jehanne; Féray, Valentin Asymptotics for skew standard Young tableaux via bounds for characters (2017) (https://arxiv.org/abs/1710.05652, to appear Proc. Amer. Math. Soc.) | Zbl

[15] Fischer, Ilse A bijective proof of the hook-length formula for shifted standard tableaux (2001) (https://arxiv.org/abs/math/0112261)

[16] Flajolet, Philippe; Sedgewick, Robert Analytic combinatorics, Cambridge University Press, Cambridge, 2009, xiv+810 pages | DOI | MR | Zbl

[17] Fomin, Sergey; Kirillov, Anatol N. Reduced words and plane partitions, J. Algebraic Combin., Volume 6 (1997) no. 4, pp. 311-319 | DOI | MR | Zbl

[18] Gorin, Vadim; Panova, Greta Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory, Ann. Probab., Volume 43 (2015) no. 6, pp. 3052-3132 | DOI | MR | Zbl

[19] Gowers, W. Timothy The two cultures of mathematics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 65-78 | MR | Zbl

[20] Greene, Curtis; Nijenhuis, Albert; Wilf, Herbert S A probabilistic proof of a formula for the number of Young tableaux of a given shape, Adv. Math., Volume 31 (1979) no. 1, pp. 104-109 | DOI | MR | Zbl

[21] Ikeda, Takeshi; Naruse, Hiroshi Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., Volume 361 (2009) no. 10, pp. 5193-5221 | DOI | MR | Zbl

[22] Ivanov, Vladimir Nikolaevich Interpolation analogues of Schur Q-functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 307 (2004) no. 10, p. 99-119, 281–282 | DOI | MR

[23] Kawanaka, Noriaki A q-series identity involving Schur functions and related topics, Osaka J. Math., Volume 36 (1999) no. 1, pp. 157-176 | MR | Zbl

[24] Kenyon, Richard Lectures on dimers, Statistical mechanics (IAS/Park City Math. Ser.), Volume 16, Amer. Math. Soc., Providence, RI, 2009, pp. 191-230 | DOI | MR | Zbl

[25] Kenyon, Richard; Okounkov, Andrei Limit shapes and the complex Burgers equation, Acta Math., Volume 199 (2007) no. 2, pp. 263-302 | DOI | MR | Zbl

[26] Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott Dimers and amoebae, Ann. of Math. (2), Volume 163 (2006) no. 3, pp. 1019-1056 | DOI | MR | Zbl

[27] Kim, Jang Soo; Oh, Suho The Selberg integral and Young books, J. Combin. Theory Ser. A, Volume 145 (2017), pp. 1-24 | DOI | MR | Zbl

[28] Kim, Jang Soo; Okada, Soichi A new q-Selberg integral, Schur functions, and Young books, Ramanujan J., Volume 42 (2017) no. 1, pp. 43-57 | DOI | MR | Zbl

[29] Kim, Jang Soo; Yoo, Meesue Product formulas for certain skew tableaux (2018) (https://arxiv.org/abs/1806.01525) | Zbl

[30] Kitaev, Sergey Patterns in permutations and words, Monographs in Theoretical Computer Science. An EATCS Series, Springer, Heidelberg, 2011, xxii+494 pages (With a foreword by Jeffrey B. Remmel) | DOI | MR | Zbl

[31] Knutson, Allen; Miller, Ezra; Yong, Alexander Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., Volume 630 (2009), pp. 1-31 | DOI | MR | Zbl

[32] Knutson, Allen; Tao, Terence Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., Volume 119 (2003) no. 2, pp. 221-260 | DOI | MR | Zbl

[33] Krattenthaler, Christian Generating functions for plane partitions of a given shape, Manuscripta Math., Volume 69 (1990) no. 2, pp. 173-201 | DOI | MR | Zbl

[34] Krattenthaler, Christian Advanced determinant calculus, Sém. Lothar. Combin., Volume 42 (1999), Paper no. B42q, 67 pages | MR | Zbl

[35] Krattenthaler, Christian Advanced determinant calculus: a complement, Linear Algebra Appl., Volume 411 (2005), pp. 68-166 | DOI | MR | Zbl

[36] Krattenthaler, Christian; Schlosser, Michael J. The major index generating function of standard Young tableaux of shapes of the form “staircase minus rectangle”, Ramanujan 125 (Contemp. Math.), Volume 627, Amer. Math. Soc., Providence, RI, 2014, pp. 111-122 | DOI | MR | Zbl

[37] Kreiman, Victor Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian (2005) (https://arxiv.org/abs/math/0512204)

[38] Kuperberg, Greg Another proof of the alternating-sign matrix conjecture, Internat. Math. Res. Notices (1996) no. 3, pp. 139-150 | DOI | MR | Zbl

[39] Kuperberg, Greg Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. (2), Volume 156 (2002) no. 3, pp. 835-866 | DOI | MR | Zbl

[40] Lai, Tri Enumeration of tilings of quasi-hexagons, hexagonal dungeons, quartered hexagons, and variants of the Aztec diamond, Ph. D. Thesis, Indiana University (USA) (2014), 217 pages | MR

[41] Lai, Tri A q-enumeration of lozenge tilings of a hexagon with three dents, Adv. in Appl. Math., Volume 82 (2017), pp. 23-57 | DOI | MR | Zbl

[42] Lakshmibai, Venkatramani; Raghavan, Komaranapuram N.; Sankaran, Parameswaran Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Q., Volume 2 (2006) no. 3, pp. 699-717 (Special Issue: In honor of Robert D. MacPherson. Part 1) | DOI | MR

[43] Lascoux, Alain Polynomials (2013) (monograph draft, http://www-igm.univ-mlv.fr/~al/ARTICLES/CoursYGKM.pdf)

[44] Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math., Volume 294 (1982) no. 13, pp. 447-450 | MR | Zbl

[45] Lascoux, Alain; Schützenberger, Marcel-Paul Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., Volume 295 (1982) no. 11, pp. 629-633 | MR | Zbl

[46] Macdonald, Ian Grant Notes on Schubert polynomials, LaCIM, UQAM, 1991

[47] Manivel, Laurent Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001, viii+167 pages (Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3) | MR | Zbl

[48] Merzon, Grigory; Smirnov, Evgeny Determinantal identities for flagged Schur and Schubert polynomials, Eur. J. Math., Volume 2 (2016) no. 1, pp. 227-245 | DOI | MR | Zbl

[49] Molev, Alexander I.; Sagan, Bruce E. A Littlewood–Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc., Volume 351 (1999) no. 11, pp. 4429-4443 | DOI | MR | Zbl

[50] Morales, Alejandro H.; Hamaker, Zach; Pak, Igor; Serrano, Luis G. (in preparation)

[51] Morales, Alejandro H.; Pak, Igor; Panova, Greta Hook formulas for skew shapes IV. Increasing tableaux and factorial Grothendieck polynomials (in preparation)

[52] Morales, Alejandro H.; Pak, Igor; Panova, Greta Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications, SIAM J. Discrete Math., Volume 31 (2017) no. 3, pp. 1953-1989 | DOI | MR | Zbl

[53] Morales, Alejandro H.; Pak, Igor; Panova, Greta Asymptotics of the number of standard Young tableaux of skew shape, European J. Combin., Volume 70 (2018), pp. 26-49 | DOI | MR | Zbl

[54] Morales, Alejandro H.; Pak, Igor; Panova, Greta Hook formulas for skew shapes I. q-analogues and bijections, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 350-405 | DOI | MR | Zbl

[55] Morales, Alejandro H.; Pak, Igor; Panova, Greta Asymptotics of principal evaluations of Schubert polynomials for layered permutations, Proc. Amer. Math. Soc., Volume 147 (2019) no. 4, pp. 1377-1389 | DOI | MR | Zbl

[56] Morales, Alejandro H.; Pak, Igor; Tassy, Martin Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings (2018) (https://arxiv.org/abs/1805.00992)

[57] Naruse, Hiroshi Schubert calculus and hook formula, 2014 (talk at 73rd Sém. Lothar. Combin., Strobl, Austria; slides available at https://www.emis.de/journals/SLC/wpapers/s73vortrag/naruse.pdf)

[58] Naruse, Hiroshi; Okada, Soichi Skew hook formula for d-complete posets, 2018 (https://arxiv.org/abs/1802.09748) | Zbl

[59] Novelli, Jean-Christophe; Pak, Igor; Stoyanovskii, Alexander V. A direct bijective proof of the hook-length formula, Discrete Math. Theor. Comput. Sci., Volume 1 (1997) no. 1, pp. 53-67 | MR | Zbl

[60] Pak, Igor Tile invariants: new horizons, Theoret. Comput. Sci., Volume 303 (2003) no. 2-3, pp. 303-331 (Tilings of the plane) | DOI | MR | Zbl

[61] Panova, Greta Tableaux and plane partitions of truncated shapes, Adv. Appl. Math., Volume 49 (2012) no. 3-5, pp. 196-217 | DOI | MR | Zbl

[62] Petrov, Leonid Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Relat. Fields, Volume 160 (2014) no. 3-4, pp. 429-487 | DOI | MR | Zbl

[63] Proctor, Robert A. New symmetric plane partition identities from invariant theory work of De Concini and Procesi, European J. Combin., Volume 11 (1990) no. 3, pp. 289-300 | DOI | MR | Zbl

[64] Romik, Dan Arctic circles, domino tilings and square Young tableaux, Ann. Probab., Volume 40 (2012) no. 2, pp. 611-647 | DOI | MR | Zbl

[65] Rota, Gian-Carlo Ten lessons I wish I had been taught, Indiscrete thoughts, Birkhäuser, 1997, pp. 195-203 | DOI

[66] Sagan, Bruce On selecting a random shifted Young tableau, J. Algorithms, Volume 1 (1980) no. 3, pp. 213-234 | DOI | MR | Zbl

[67] Sagan, Bruce The symmetric group, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages (Representations, combinatorial algorithms, and symmetric functions) | DOI | MR | Zbl

[68] Sage-Combinat community Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2017 (https://wiki.sagemath.org/combinat)

[69] Serrano, Luis G.; Stump, Christian Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials, Electron. J. Combin., Volume 19 (2012) no. 1, Paper no. P16, 18 pages | MR | Zbl

[70] Sloane, Neil James Alexander The Online Encyclopedia of Integer Sequences (https://oeis.org/)

[71] Stanley, Richard P. Plane partitions: past, present, and future, Combinatorial Mathematics: Proceedings of the Third International Conference (New York, 1985) (Ann. New York Acad. Sci.), Volume 555, New York Acad. Sci., New York, 1989, pp. 397-401 | DOI | MR | Zbl

[72] Stanley, Richard P. Enumerative Combinatorics, 2, Cambridge University Press, 1999 | Zbl

[73] Stanley, Richard P. A survey of alternating permutations, Combinatorics and graphs (Contemp. Math.), Volume 531, Amer. Math. Soc., Providence, RI, 2010, pp. 165-196 | DOI | MR | Zbl

[74] Stanley, Richard P. Enumerative Combinatorics, 1, Cambridge University Press, 2012 | Zbl

[75] Stanley, Richard P. Some Schubert shenanigans (2017) (https://arxiv.org/abs/1704.00851)

[76] Stembridge, John R. Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math., Volume 83 (1990) no. 1, pp. 96-131 | DOI | MR | Zbl

[77] Thurston, William P. Groups, tilings and finite state automata, Geometry Computing Group, 1989

[78] Wachs, Michelle L. Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory Ser. A, Volume 40 (1985) no. 2, pp. 276-289 | DOI | MR | Zbl

[79] Weigandt, Anna E. Schubert polynomials, 132-patterns, and Stanley’s conjecture, Algebr. Comb., Volume 1 (2018) no. 4, pp. 415-423 | MR | Zbl

[80] Woo, Alexander Catalan numbers and Schubert polynomials for w=1(n+1)...2 (2004) (https://arxiv.org/abs/math/0407160)

Cited by Sources: