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Bijections and the Riordan group. (English) Zbl 1048.05008
Summary: One of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space. We examine combinatorial sequences in the context of the Riordan group. Various subgroups of the Riordan group each give us a different view of the original sequence. In many cases this leads to both a combinatorial interpretation and to ECO rewriting rules. We concentrate on just four of the subgroups of the Riordan group to demonstrate some of the possibilities of this approach.

MSC:
05A15 Exact enumeration problems, generating functions
Software:
OEIS
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References:
[1] Aigner, M., Motzkin numbers, European J. combin., 19, 663-675, (1999) · Zbl 0915.05004
[2] E. Barcucci, A Del Lungo, A. Frosini, S. Rinaldi, From rational functions to regular languages, Proc. 12th FPSAC, Moscow, 2000. · Zbl 0960.68100
[3] Barcucci, E.; Del Lungo, A; Pergola, E.; Pinzani, R., ECO, A methodology for the enumeration of combinatorial objects, J. difference equations appl., 5, 40-52, (1999) · Zbl 0934.05008
[4] Barcucci, E.; Pergola, E.; Pinzani, R.; Rinaldi, S., ECO-systems for Dyck and Schröder paths, Pu.m.a., 11, 401-407, (2000) · Zbl 0980.05004
[5] Barcucci, E.; Rinaldi, S., Some linear recurrences and their combinatorial interpretations by means of regular languages, Theoret. comput. sci., 255, 679-686, (2001) · Zbl 0974.68101
[6] Deutsch, E.; Shapiro, L., A survey of the fine numbers, Discrete math., 241, 241-265, (2001) · Zbl 0992.05011
[7] D. Knuth, The Art of Computer Programming, Vol. 1, Addison-Wesley, Reading, MA, 1973. · Zbl 0302.68010
[8] C. Mallows, L. Shapiro, Balls on the lawn, J. Integer Sequences (electronic, www.research.att.com/ njas/sequences/JIS/) article 99.1.5. · Zbl 0923.05004
[9] Merlini, D.; Rogers, D.; Sprugnoli, R.; Verri, M.C., The tennis ball problem, J. combin. theory, ser. A, 99, 73-88, (2002) · Zbl 1006.05007
[10] Shapiro, L.W.; Getu, S.; Woan, W.-J.; Woodson, L., The Riordan group, Discrete appl. math., 34, 229-239, (1991) · Zbl 0754.05010
[11] N.J.A. Sloane, S. Plouœe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995. Also on line at www.research.att.com/ njas/sequences/index..
[12] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete math., 132, 267-290, (1994) · Zbl 0814.05003
[13] R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0928.05001
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