zbMATH — the first resource for mathematics

Hook formulas for skew shapes. I: \(q\)-analogues and bijections. (English) Zbl 1373.05026
Summary: The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In [“Schubert calculus and hook formula”, talk at the 73rd séminaire lotharingien de combinatoire, Strobl (Austria), 8–10 September 2014, http://tinyurl.com/z6paqzu], H. Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functions and a generalization of the Hillman-Grassl correspondence, respectively.
The main new results are two different \(q\)-analogues of Naruse’s formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. We establish explicit bijections between these objects and families of integer arrays with certain nonzero entries, which also proves the second formula.

05A30 \(q\)-calculus and related topics
05E10 Combinatorial aspects of representation theory
Full Text: DOI
[1] Adin, R.; Roichman, Y., Standard Young tableaux, (Bóna, M., Handbook of Enumerative Combinatorics, (2015), CRC Press Boca Raton), 895-974 · Zbl 1330.05162
[2] Andersen, H. H.; Jantzen, J. C.; Soergel, W., Representations of quantum groups at p-th root of unity and of semisimple groups in characteristic p: independence of p, Astérisque, 220, 321, (1994) · Zbl 0802.17009
[3] Bandlow, J., An elementary proof of the hook formula, Electron. J. Combin., 15, (2008), RP 45, 14 pp. · Zbl 1179.05118
[4] Billey, S., Kostant polynomials and the cohomology ring for \(G / B\), Duke Math. J., 96, 205-224, (1999) · Zbl 0980.22018
[5] Borodin, A.; Gorin, V.; Rains, E. M., q-distributions on boxed plane partitions, Selecta Math., 16, 731-789, (2010) · Zbl 1205.82122
[6] Chen, X.; Stanley, R. P., A formula for the specialization of skew Schur functions, Ann. Comb., 20, 539-548, (2017) · Zbl 1347.05248
[7] Ciocan-Fontanine, I.; Konvalinka, M.; Pak, I., The weighted hook length formula, J. Combin. Theory Ser. A, 118, 1703-1717, (2011) · Zbl 1227.05034
[8] Fischer, I., A bijective proof of the hook-length formula for shifted standard tableaux
[9] Frame, J. S.; Robinson, G. de B.; Thrall, R. M., The hook graphs of the symmetric group, Canad. J. Math., 6, 316-324, (1954) · Zbl 0055.25404
[10] Galashin, P., A Littlewood-Richardson rule for dual stable Grothendieck polynomials · Zbl 1366.05116
[11] Gansner, E. R., The hillman-grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory Ser. A, 30, 71-89, (1981) · Zbl 0474.05008
[12] Gansner, E. R., Matrix correspondences of plane partitions, Pacific J. Math., 92, 295-315, (1981) · Zbl 0432.05010
[13] I.M. Gessel, X.G. Viennot, Determinants, paths and plane partitions, preprint, 1989; available at tinyurl.com/zv3wvyh.
[14] Graham, W.; Kreiman, V., Excited Young diagrams, equivariant K-theory, and Schubert varieties, Trans. Amer. Math. Soc., 367, 6597-6645, (2015) · Zbl 1317.05187
[15] Greene, C.; Nijenhuis, A.; Wilf, H. S., A probabilistic proof of a formula for the number of Young tableaux of a given shape, Adv. Math., 31, 104-109, (1979) · Zbl 0398.05008
[16] Hillman, A. P.; Grassl, R. M., Reverse plane partitions and tableau hook numbers, J. Combin. Theory Ser. A, 21, 216-221, (1976) · Zbl 0341.05008
[17] Huber, M., Fast perfect sampling from linear extensions, Discrete Math., 306, 420-428, (2006) · Zbl 1090.60064
[18] Ikeda, T.; Naruse, H., Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., 361, 5193-5221, (2009) · Zbl 1229.05287
[19] Ikeda, T.; Naruse, H., K-theoretic analogues of factorial Schur P- and Q-functions, Adv. Math., 243, 22-66, (2012) · Zbl 1278.05240
[20] Knutson, A., A Schubert calculus recurrence from the noncomplex W-action on \(G / B\)
[21] Knutson, A.; Miller, E.; Yong, A., Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., 630, 1-31, (2009) · Zbl 1169.14033
[22] Knutson, A.; Tao, T., Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., 119, 221-260, (2003) · Zbl 1064.14063
[23] Konvalinka, M., A bijective proof of the hook-length formula for skew shapes · Zbl 1379.05123
[24] Kostant, B.; Kumar, S., The nil Hecke ring and cohomology of \(G / P\) for Kac-Moody group, Adv. Math., 62, 183-237, (1986) · Zbl 0641.17008
[25] Krattenthaler, C., Bijective proofs of the hook formulas for the number of standard Young tableaux, ordinary and shifted, Electron. J. Combin., 2, (1995), RP 13, 9 pp. · Zbl 0822.05065
[26] Krattenthaler, C., An involution principle-free bijective proof of Stanley’s hook-content formula, Discrete Math. Theor. Comput. Sci., 3, 11-32, (1998) · Zbl 0934.05124
[27] Krattenthaler, C., Another involution principle-free bijective proof of Stanley’s hook-content formula, J. Combin. Theory Ser. A, 88, 66-92, (1999) · Zbl 0936.05087
[28] Krattenthaler, C., Plane partitions in the work of richard Stanley and his school · Zbl 1365.05016
[29] Kreiman, V., Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian
[30] Kreiman, V., Schubert classes in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian
[31] Lam, T.; Pylyavskyy, P., Combinatorial Hopf algebras and K-homology of Grassmannians, Int. Math. Res. Not. IMRN, 2007, 24, 1-48, (2007) · Zbl 1134.16017
[32] Lascoux, A.; Schützenberger, M.-P., Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math., 294, 447-450, (1982), (in French) · Zbl 0495.14031
[33] Molev, A. I.; Sagan, B. E., A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc., 351, 4429-4443, (1999) · Zbl 0972.05053
[34] Morales, A. H.; Pak, I.; Panova, G., Hook formulas for skew shapes · Zbl 1373.05026
[35] Morales, A. H.; Pak, I.; Panova, G., Hook formulas for skew shapes II. combinatorial proofs and enumerative applications, SIAM J. Discrete Math., 31, 1953-1989, (2017) · Zbl 1370.05007
[36] Morales, A. H.; Pak, I.; Panova, G., Hook formulas for skew shapes III. multivariate and product formulas · Zbl 1373.05026
[37] A.H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes IV. Increasing tableaux and factorial Grothendieck polynomials, in preparation. · Zbl 1373.05026
[38] Morales, A. H.; Pak, I.; Panova, G., Asymptotics for the number of standard Young tableaux of skew shape · Zbl 1384.05175
[39] Narayanan, H., On the complexity of computing kostka numbers and Littlewood-Richardson coefficients, J. Algebraic Combin., 24, 347-354, (2006) · Zbl 1101.05066
[40] H. Naruse, Schubert calculus and hook formula, talk slides at 73rd Sém. Lothar. Combin, Strobl, Austria, 2014; available at tinyurl.com/z6paqzu.
[41] Novelli, J.-C.; Pak, I.; Stoyanovskii, A. V., A direct bijective proof of the hook-length formula, Discrete Math. Theor. Comput. Sci., 1, 53-67, (1997) · Zbl 0934.05125
[42] Okounkov, A.; Olshanski, G., Shifted Schur functions, St. Petersburg Math. J., 9, 239-300, (1998)
[43] Pak, I., Hook length formula and geometric combinatorics, Sém. Lothar. Combin., 46, (2001), Art. B46f, 13 pp. · Zbl 0982.05109
[44] Remmel, J. B., Bijective proofs of formulae for the number of standard Young tableaux, Linear Multilinear Algebra, 11, 45-100, (1982) · Zbl 0485.05005
[45] Sagan, B. E., Enumeration of partitions with hooklengths, European J. Combin., 3, 85-94, (1982) · Zbl 0483.05010
[46] Sagan, B. E., The symmetric group, (2000), Springer
[47] Sage-combinat: enhancing sage as a toolbox for computer exploration in algebraic combinatorics, (2013)
[48] Stanley, R. P., Theory and applications of plane partitions, part 2, Stud. Appl. Math., 50, 259-279, (1971) · Zbl 0225.05012
[49] Stanley, R. P., The conjugate trace and trace of a plane partition, J. Combin. Theory Ser. A, 14, 53-65, (1973) · Zbl 0251.05006
[50] Stanley, R. P.; Stanley, R. P., Enumerative combinatorics, vol. 2, (1999), Cambridge Univ. Press · Zbl 0928.05001
[51] Stembridge, J. R., On the fully commutative elements of Coxeter groups, J. Algebraic Combin., 5, 353-385, (1996) · Zbl 0864.20025
[52] Thomas, H.; Yong, A., Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier, (2017), in press
[53] Tymoczko, J. S., Billey’s formula in combinatorics, geometry, and topology, Math. Stat.: Fac. Publ., Smith Coll., 10, (2013) · Zbl 1379.05125
[54] Vershik, A. M., The hook formula and related identities, J. Sov. Math., 59, 1029-1040, (1992)
[55] Viennot, G., Une forme géométrique de la correspondance de Robinson-Schensted, Lecture Notes in Math., vol. 579, 29-58, (1977), Springer Berlin · Zbl 0389.05016
[56] Wachs, M., Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory Ser. A, 40, 276-289, (1985) · Zbl 0579.05001
[57] White, D. E., Some connections between the Littlewood-Richardson rule and the construction of Schensted, J. Combin. Theory Ser. A, 30, 237-247, (1981) · Zbl 0472.05005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.