×

Some determinants of path generating functions. (English) Zbl 1227.05067

Summary: We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers-most notably of Catalan, ballot, and of Motzkin numbers-that appeared previously in the literature.

MSC:

05A19 Combinatorial identities, bijective combinatorics
05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions
11C20 Matrices, determinants in number theory
15A15 Determinants, permanents, traces, other special matrix functions

Software:

DODGSON; DET
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aigner, M., Catalan and other numbers: a recurrent theme, (Crapo, H.; Senato, D., Algebraic Combinatorics and Computer Science (2001), Springer-Verlag: Springer-Verlag Berlin), 347-390 · Zbl 0971.05002
[2] Amdeberhan, T.; Zeilberger, D., Determinants through the looking glass, Adv. in Appl. Math., 27, 225-230 (2001), Maple package DODGSON available at · Zbl 0994.05018
[3] Bressoud, D. M., Proofs and Confirmations — The Story of the Alternating Sign Matrix Conjecture (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0944.05001
[4] Comtet, L., Advanced Combinatorics (1974), D. Reidel: D. Reidel Dordrecht, Holland
[5] Eğecioğlu, Ö.; Redmond, T.; Ryavec, C., Almost product evaluation of Hankel determinants, Electron. J. Combin., 15, 1 (2008), Article #R6, 58 pp · Zbl 1206.05009
[6] Eğecioğlu, Ö.; Redmond, T.; Ryavec, C., A multilinear operator for almost product evaluation of Hankel determinants, J. Combin. Theory Ser. A, 117, 77-103 (2010) · Zbl 1227.05031
[7] Fulmek, M.; Krattenthaler, C., Lattice path proofs for determinant formulas for symplectic and orthogonal characters, J. Combin. Theory Ser. A, 77, 3-50 (1997) · Zbl 0867.05083
[8] Fulton, W.; Harris, J., Representation Theory (1991), Springer-Verlag: Springer-Verlag New York
[9] Gessel, I. M.; Viennot, X., Binomial determinants, paths, and hook length formulae, Adv. Math., 58, 300-321 (1985) · Zbl 0579.05004
[10] Gessel, I. M.; Viennot, X., Determinants, paths, and plane partitions (1989), preprint, available at
[11] Ghorpade, S. R.; Krattenthaler, C., The Hilbert series of Pfaffian rings, (Christensen, C.; Sundaram, G.; Sathaye, A.; Bajaj, C., Algebra, Arithmetic and Geometry with Applications (2004), Springer-Verlag: Springer-Verlag New York), 337-356 · Zbl 1083.13504
[12] Krattenthaler, C., The enumeration of lattice paths with respect to their number of turns, (Balakrishnan, N., Advances in Combinatorial Methods and Applications to Probability and Statistics (1997), Birkhäuser: Birkhäuser Boston), 29-58 · Zbl 0882.05004
[13] Krattenthaler, C., Advanced determinant calculus, Sem. Lothar. Combin., 42 (1999), (“The Andrews Festschrift”), Article B42q, 67 pp · Zbl 0923.05007
[14] Krattenthaler, C., Advanced determinant calculus: a complement, Linear Algebra Appl., 411, 64-166 (2005) · Zbl 1079.05008
[15] Krattenthaler, C., On multiplicities of points on Schubert varieties in Graßmannians, II, J. Algebraic Combin., 22, 273-288 (2005) · Zbl 1093.14068
[16] Krattenthaler, C., Watermelon configurations with wall interaction: exact and asymptotic results, J. Phys. Conf. Ser., 42, 179-212 (2006)
[17] Lindström, B., On the vector representations of induced matroids, Bull. Lond. Math. Soc., 5, 85-90 (1973) · Zbl 0262.05018
[18] Stanley, R. P., Enumerative Combinatorics, vol. 2 (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0928.05001
[19] Stembridge, J. R., Nonintersecting paths, pfaffians and plane partitions, Adv. Math., 83, 96-131 (1990) · Zbl 0790.05007
[20] Viennot, X., Une théorie combinatoire des polynômes orthogonaux généraux (1983), UQAM: UQAM Montréal, Québec
[21] Zeilberger, D., The holonomic ansatz, II. Automatic discovery (!) and proof (!!) of holonomic determinant evaluations, Ann. Comb., 11, 241-247 (2007), Maple package DET available at · Zbl 1125.05009
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.