×

Difference operators for partitions under the Littlewood decomposition. (English) Zbl 1377.05013

Summary: The concept of \(t\)-difference operator for functions of partitions is introduced to prove a generalization of Stanley’s theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of \(t\)-hooks. It is well known that the hook lengths of multiples of \(t\) can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo \(t\).

MSC:

05A17 Combinatorial aspects of partitions of integers
05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
11P81 Elementary theory of partitions
47B39 Linear difference operators
39A70 Difference operators
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Amdeberhan, T.: Differential operators, shifted parts, and hook lengths. Ramanujan J. 24(3), 259-271 (2011) · Zbl 1234.05232 · doi:10.1007/s11139-010-9271-0
[2] Armstrong, D., Hanusa, C., Jones, B.: Results and conjectures on simultaneous core partitions. Eur. J. Combin. 41, 205-220 (2014) · Zbl 1297.05024 · doi:10.1016/j.ejc.2014.04.007
[3] Bandlow, J.: An elementary proof of the hook formula. Electron. J. Combin. 15, research paper 45 (2008) · Zbl 1179.05118
[4] Carde, K., Loubert, J., Potechin, A., Sanborn, A.: Proof of Han’s Hook expansion conjecture. Preprint. arXiv:0808.0928 · Zbl 0398.05008
[5] Frame, J.S., Robinson, G. de B., Thrall, R.M.: The hook graphs of \[S_nSn\]. Can. J. Math. 6, 316-324 (1954) · Zbl 0055.25404
[6] 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(1), 104-109 (1979) · Zbl 0398.05008 · doi:10.1016/0001-8708(79)90023-9
[7] Greene, C., Nijenhuis, A., Wilf, H.S.: Another probabilistic method in the theory of Young tableaux. J. Combin. Theory Ser. A 37(2), 127-135 (1984) · Zbl 0561.05004 · doi:10.1016/0097-3165(84)90065-7
[8] Han, G.-N.: Some conjectures and open problems on partition hook lengths. Exp. Math. 18, 97-106 (2009) · Zbl 1167.05004 · doi:10.1080/10586458.2009.10128888
[9] Han, G.-N.: The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension, and applications. Ann. Inst. Fourier 60(1), 1-29 (2010) · Zbl 1215.05013 · doi:10.5802/aif.2515
[10] Han, G.-N.: Hook lengths and shifted parts of partitions. Ramanujan J. 23(1-3), 127-135 (2010) · Zbl 1218.05014 · doi:10.1007/s11139-009-9170-4
[11] Han, G.-N., Xiong, H.: Difference operators for partitions and some applications. Preprint. arXiv:1508.00772 · Zbl 1395.05014
[12] Han, G.-N., Xiong, H.: New hook-content formulas for strict partitions. Preprint. arXiv:1511.02829 · Zbl 1373.05009
[13] Han, G.-N., Xiong, H.: Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions. Preprint. arXiv:1601.04369 · Zbl 1421.05015
[14] James, G.; Kerber, A.; Rota, G-C (ed.), The representation theory of the symmetric group, No. 16 (1981), Reading · Zbl 0491.20010
[15] Knuth, D.: The Art of Computer Programming, vol. 3: Sorting and Searching. Addison-Wesley, London (1973) · Zbl 0302.68010
[16] Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press/Oxford University Press, New York (1995) · Zbl 0824.05059
[17] Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions. In: Progress in Mathematics, vol. 244, pp. 525-596. Birkhäuser, Boston (2006) · Zbl 1233.14029
[18] Olshanski, G.: Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter. Int. Math. Res. Not. (IMRN) 6, 1102-1166 (2010) · Zbl 1193.60097
[19] Olshanski, G.: Plancherel averages: remarks on a paper by Stanley. Electron. J. Combin. 17, research paper 43 (2010) · Zbl 1193.05161
[20] Panova, G.: Polynomiality of some hook-length statistics. Ramanujan J. 27(3), 349-356 (2012) · Zbl 1244.05230 · doi:10.1007/s11139-011-9332-z
[21] Rota, G.-C.: On the foundations of combinatorial theory: I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie Verw. Geb. 2, 349-356 (1964)
[22] Stanley, R.P.: Differential posets. J. Am. Math. Soc. 1(4), 919-961 (1988) · Zbl 0658.05006 · doi:10.1090/S0894-0347-1988-0941434-9
[23] Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, New York (1999) · Zbl 0928.05001 · doi:10.1017/CBO9780511609589
[24] Stanley, R.P.: Some combinatorial properties of hook lengths, contents, and parts of partitions. Ramanujan J. 23(1-3), 91-105 (2010) · Zbl 1234.05234 · doi:10.1007/s11139-009-9185-x
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.