“Motzkin Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Motzkin Number

释义

Motzkin Number

The Motzkin numbers enumerate various combinatorial objects. Donaghey and Shapiro (1977) give 14 different manifestations ofthese numbers. In particular, they give the number of paths from (0, 0) to (, 0) which never dip below and are madeup only of the steps (1, 0), (1, 1), and (1, ), i.e., , , and . The first are 1, 2, 4,9, 21, 51, ... (Sloane's A001006). The Motzkin number Generating Function satisfies

(1)

and is given by

(2)

or by the Recurrence Relation

(3)

with

. The Motzkin number

is also given by

			(4)

			(5)

where

is a Binomial Coefficient.

See also Catalan Number, King Walk, Schröder Number

References

Barcucci, E.; Pinzani, R.; and Sprugnoli, R. ``The Motzkin Family.'' Pure Math. Appl. Ser. A 2, 249-279, 1991.

Donaghey, R. ``Restricted Plane Tree Representations of Four Motzkin-Catalan Equations.'' J. Combin. Th. Ser. B 22, 114-121, 1977.

Donaghey, R. and Shapiro, L. W. ``Motzkin Numbers.'' J. Combin. Th. Ser. A 23, 291-301, 1977.

Kuznetsov, A.; Pak, I.; and Postnikov, A. ``Trees Associated with the Motzkin Numbers.'' J. Combin. Th. Ser. A 76, 145-147, 1996.

Motzkin, T. ``Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula for Partitions of a Polygon, for Permanent Preponderance, and for Nonassociative Products.'' Bull. Amer. Math. Soc. 54, 352-360, 1948.

Sloane, N. J. A. SequenceA001006/M1184in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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