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Combinatorics of Riordan arrays with identical \(A\) and \(Z\) sequences. (English) Zbl 1243.05007
Summary: In theory, Riordan arrays can have any A-sequence and any Z-sequence. For examples of combinatorial interest they tend to be related. Here we look at the case that they are identical or nearly so. We provide a combinatorial interpretation in terms of weighted Łukasiewicz paths and then look at several large classes of examples.

MSC:
05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
Software:
OEIS
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[1] Deutsch, E., Ordered trees with prescribed root degrees, node degrees, and branch lengths, Discrete math., 282, 89-94, (2004) · Zbl 1042.05026
[2] Deutsch, E.; Shapiro, L.W., A survey of the fine numbers, Discrete math., 241, 241-265, (2001) · Zbl 0992.05011
[3] Gessel, I.M., A factorization for formal Laurent series and lattice path enumeration, J. combinatorial theory, series A, 28, 321-337, (1980)
[4] Graham, R.; Knuth, D.; Patashnik, O., Concrete mathematics, (1994), Addison-Wesley Pub. Co.
[5] He, T.-X.; Sprugnoli, R., Sequence characterization of Riordan arrays, Discrete math., 309, 3962-3974, (2009) · Zbl 1228.05014
[6] Merlini, D.; Rogers, D.G.; Sprugnoli, R.; Verri, M.C., On some alterative characterizations of Riordan arrays, Can. J. math., 49, 301-320, (1997) · Zbl 0886.05013
[7] Merlini, D.; Sprugnoli, R., The relevant prefixes of coloured Motzkin walks: an average case analysis, Theoret. comput. sci., 411, 148-163, (2010) · Zbl 1189.68181
[8] Merlini, D.; Verri, M.C., Generating trees and proper Riordan arrays, Discrete math., 218, 167-183, (2000) · Zbl 0949.05004
[9] Pergola, E.; Pinzani, R.; Rinaldi, S.; Sulanke, R.A., A bijective approach to the area of generalized Motzkin paths, Adv. appl. math., 28, 580-591, (2002) · Zbl 1005.05008
[10] Rogers, D.G., A Schröder triangle: three combinatorial problems, (), 175-196 · Zbl 0368.05004
[11] Shapiro, L.W.; Getu, S.; Woan, W.-J.; Woodson, L., The Riordan group, Discrete appl. math., 34, 229-239, (1991) · Zbl 0754.05010
[12] Shapiro, L.W.; Wang, C.J., A bijection between 3-Motzkin paths and Schröder paths with no peak at odd height, J. integer seq., 12, (2009), Article 09.3.2 · Zbl 1165.05300
[13] Sloane, N.J.A., The on-line encyclopedia of integer sequences · Zbl 1274.11001
[14] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete math., 132, 267-290, (1994) · Zbl 0814.05003
[15] Stanley, R.P., Enumerative combinatorics, vol. 2, (1997), Cambridge Univ. Press · Zbl 0889.05001
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