Strict unimodality of q-binomial coefficients
Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 415-418.

We prove the strict unimodality of the q-binomial coefficients (nk)q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of Sn representations.

Nous prouvons lʼunimodalité stricte des coefficients q-binomiaux (nk)q comme polynômes en q. La preuve est basée sur la combinatoire de certains tableaux de Young et la propriété du semi-groupe des coefficients de Kronecker des représentations de Sn.

Published online:
DOI: 10.1016/j.crma.2013.06.008
Pak, Igor 1; Panova, Greta 1

1 Department of Mathematics, UCLA, Los Angeles, CA 90095, USA
     author = {Pak, Igor and Panova, Greta},
     title = {Strict unimodality of \protect\emph{q}-binomial coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--418},
     publisher = {Elsevier},
     volume = {351},
     number = {11-12},
     year = {2013},
     doi = {10.1016/j.crma.2013.06.008},
     language = {en},
     url = {}
AU  - Pak, Igor
AU  - Panova, Greta
TI  - Strict unimodality of q-binomial coefficients
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 415
EP  - 418
VL  - 351
IS  - 11-12
PB  - Elsevier
UR  -
DO  - 10.1016/j.crma.2013.06.008
LA  - en
ID  - CRMATH_2013__351_11-12_415_0
ER  - 
%0 Journal Article
%A Pak, Igor
%A Panova, Greta
%T Strict unimodality of q-binomial coefficients
%J Comptes Rendus. Mathématique
%D 2013
%P 415-418
%V 351
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2013.06.008
%G en
%F CRMATH_2013__351_11-12_415_0
Pak, Igor; Panova, Greta. Strict unimodality of q-binomial coefficients. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 415-418. doi : 10.1016/j.crma.2013.06.008.

[1] Brenti, F. (Mem. Am. Math. Soc.), Volume vol. 413 (1989), p. 106

[2] Brenti, F. Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Contemp. Math., vol. 178, AMS, Providence, RI, 1994, pp. 71-89

[3] Christandl, M.; Harrow, A.W.; Mitchison, G. Nonzero Kronecker coefficients and what they tell us about spectra, Commun. Math. Phys., Volume 270 (2007), pp. 575-585

[4] Kirillov, A.N. Unimodality of generalized Gaussian coefficients, C. R. Acad. Sci. Paris, Ser. I, Volume 315 (1992) no. 5, pp. 497-501

[5] Kirillov, A.N. An invitation to the generalized saturation conjecture, Publ. RIMS, Volume 40 (2004), pp. 1147-1239

[6] Lindström, B. A partition of L(3,n) into saturated symmetric chains, Eur. J. Comb., Volume 1 (1980), pp. 61-63

[7] Macdonald, I.G. An elementary proof of a q-binomial identity, q-Series and Partitions, Inst. Math. and Its Appl., vol. 18, Springer, New York, 1989, pp. 73-75

[8] Macdonald, I.G. Symmetric Functions and Hall Polynomials, Oxford University Press, New York, 1995

[9] Manivel, L. On rectangular Kronecker coefficients, J. Algebr. Comb., Volume 33 (2011), pp. 153-162

[10] Mizukawa, H.; Yamada, H.-F. Rectangular Schur functions and the basic representation of affine Lie algebras, Discrete Math., Volume 298 (2005), pp. 285-300

[11] OʼHara, K.M. Unimodality of Gaussian coefficients: a constructive proof, J. Comb. Theory, Ser. A, Volume 53 (1990), pp. 29-52

[12] Pak, I.; Panova, G. Unimodality via Kronecker products | arXiv

[13] Pak, I.; Panova, G.; Vallejo, E. Kronecker products, characters, partitions, and the tensor square conjectures | arXiv

[14] Reid, M. Klarner systems and tiling boxes with polyominoes, J. Comb. Theory, Ser. A, Volume 111 (2005), pp. 89-105

[15] Stanley, R.P. Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. N.Y. Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500-535

[16] Stanley, R.P. Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999

[17] Sylvester, J.J. Proof of the hitherto undemonstrated Fundamental Theorem of Invariants, Philos. Mag. (Coll. Math. Papers), Volume 5 (1878), pp. 178-188 (reprinted, vol. 3, 1973, pp. 117-126 available at)

[18] E. Vallejo, Kronecker squares of complex Sn characters and Littlewood–Richardson multi-tableaux, preprint.

[19] West, D.B. A symmetric chain decomposition of L(4,n), Eur. J. Comb., Volume 1 (1980), pp. 379-383

[20] Zeilberger, D. Kathy OʼHaraʼs constructive proof of the unimodality of the Gaussian polynomials, Am. Math. Mon., Volume 96 (1989), pp. 590-602

[21] Zelevinsky, A. Littlewood–Richardson semigroups, New Perspectives in Algebraic Combinatorics, Cambridge University Press, Cambridge, 1999, pp. 337-345

Cited by Sources: