The Riordan group

A method for the enumeration of various classes of column-convex polygons

In this paper Motzkin numbersMn(which are related to Catalan numbers) are studied. The (known) connection to Tchebychev polynomials is discussed with applications to the Hankel matrices of Motzkin numbers. It is shown that the sequenceMnis logarithmically concave with limMn+1/Mn=3. Finally, two ballot-number type sequences forMnare derived, with an application to directed animals.

Motzkin Numbers

These are partitions of[l]={1, 2, . . . , l}intonblocks such that no four-term subsequence of [l]induces the mentioned pattern and eachkconsecutive numbers of[l]fall into different blocks. These structures are motivated by Davenport–Schinzel sequences. We summarize and extend known enumeriative results for the patternp=ababand give an explicit formula for the numberp(abab, n, l, k)of such partitions. Our main tools are generating functions. We determine the corresponding generating function forp=abbaandk=1, 2, 3.Fork=2there is a connection with the number of directed animals. We solve exactly two related extremal problems.

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D Gouyou-Beauchamps

Papers

Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem