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Determinants involving \(q\)-Stirling numbers. (English) Zbl 1071.05011
Summary: Let \(S[i,j]\) denote the \(q\)-Stirling numbers of the second kind. We show that the determinant of the matrix \((S[s+ i+ j,s+ j])_{0\leq i,j\leq n}\) is given by the product \(q^{{s+n+1\choose 3}-{s\choose 3}}\cdot[s]^0\cdot [s+1]^1\cdots[s+ n]^n\). We give two proofs of this result, one bijective and one based upon factoring the matrix. We also prove an identity due to J. Cigler [Sitzungsber., Abt. II, Österr. Akad. Wiss. Math.-Naturwiss. Kl. 208, 143–157 (1999; Zbl 1023.33011)] that expresses the Hankel determinant of \(q\)-exponential polynomials as a product. Lastly, a two variable version of a theorem of Sylvester and an application are presented.

MSC:
05A30 \(q\)-calculus and related topics
05A10 Factorials, binomial coefficients, combinatorial functions
11B73 Bell and Stirling numbers
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[1] Aigner, M., A characterization of the Bell numbers, Discrete math., 205, 207-210, (1999) · Zbl 0959.11014
[2] Buhler, J.; Eisenbud, D.; Graham, R.; Wright, C., Juggling drops and descents, Amer. math. monthly, 101, 507-519, (1994) · Zbl 0814.05002
[3] Buhler, J.; Graham, R., A note on the binomial drop polynomial of a poset, J. combin. theory ser. A, 66, 321-326, (1994) · Zbl 0797.06002
[4] Carlitz, L., q-Bernoulli numbers and polynomials, Duke math. J., 15, 987-1000, (1948) · Zbl 0032.00304
[5] Cigler, J., Eine charakterisierung der q-exponentialpolynome, Österreich. akad. wiss. math.-natur. kl. sitzungsber. II, 208, 143-157, (1999) · Zbl 1023.33011
[6] de Médicis, A.; Leroux, P., A unified combinatorial approach for q- (and p,q-) Stirling numbers, J. statist. plann. inference, 34, 89-105, (1993) · Zbl 0783.05005
[7] Ehrenborg, R., The Hankel determinant of exponential polynomials, Amer. math. monthly, 107, 557-560, (2000) · Zbl 0985.15006
[8] Ehrenborg, R.; Readdy, M., Juggling and applications to q-analogues, Discrete math., 157, 107-125, (1996) · Zbl 0859.05010
[9] Garsia, A.M.; Remmel, J.B., q-counting rook configurations and a formula of Frobenius, J. combin. theory ser. A, 41, 246-275, (1986) · Zbl 0598.05007
[10] Goldman, J.; Rota, G.-C., On the foundations of combinatorial theory. IV. finite vector spaces and Eulerian generating functions, Stud. appl. math., 49, 239-258, (1970) · Zbl 0212.03303
[11] Labelle, G.; Leroux, P.; Pergola, E.; Pinzani, R., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete math., 246, 177-195, (2002) · Zbl 1001.05007
[12] Milne, S.C., A q-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials, Trans. amer. math. soc., 245, 89-118, (1978) · Zbl 0402.05007
[13] Milne, S.C., Restricted growth functions, rank row matchings of partition lattices, and q-Stirling numbers, Adv. math., 43, 173-196, (1982) · Zbl 0482.05012
[14] Radoux, C., Calcul effectif de certains déterminants de Hankel, Bull. soc. math. belg. Sér. B, 31, 49-55, (1979) · Zbl 0439.10006
[15] Radoux, C., Déterminant de Hankel construit sur des polynomes liés aux nombres de dérangements, European J. combin., 12, 327-329, (1991) · Zbl 0805.05003
[16] Radoux, C., Déterminants de Hankel et théorème de Sylvester, (), 115-122
[17] Radoux, C., Addition formulas for polynomials built on classical combinatorial sequences, J. comput. appl. math., 115, 471-477, (2000) · Zbl 1025.11004
[18] Radoux, C., The Hankel determinant of exponential polynomials: a very short proof and a new result concerning Euler numbers, Amer. math. monthly, 109, 277-278, (2002) · Zbl 1056.11012
[19] Sagan, B.E., A maj statistic for partitions, European J. combin., 12, 69-79, (1991) · Zbl 0728.05007
[20] Wachs, M.; White, D., p,q-Stirling numbers and set partition statistics, J. combin. theory ser. A, 56, 27-46, (1991) · Zbl 0732.05004
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