VARIANTS OF SCHROEDER DISSECTIONS

Leonard M. Smiley

Department of Mathematical Sciences, University of Alaska Anchorage, Anchorage, Alaska 99508

Abstract. Some formulae are given for the enumeration of certain types of dissections of the convex (n + 2)-gon by non-crossing diagonals. A cataloguing tool, the reversive symbol, is proposed.

July 5, 1999

1. Dissections and Reversions

When m diagonals are drawn in a convex (n+2)-gon, (0 ≤ m ≤ n−1), so that no two diagonals intersect, we call this a dissection, and the resulting diagonal-free polygon(s) the tiles of the dissection. If the vertices of the polygon are labelled, we may ask how many distinct dissections of the (n + 2)-gon are possible. The answer is an integer sequence known as the solution to Schroeder’s Second Problem, or the Schroeder numbers. If in addition we require that m = n − 1, or, equivalently, that all tiles are triangles, or that there are exactly n tiles, the solution is called the sequence of Catalan numbers. Variant restrictions have been placed on Schroeder dissections and enumerations achieved by a variety of methods (see references in [S1]). A solution may be given by a closed formula or a summation formula for the nth term; a recurrence relation defining the sequence; or a closed formula for the generating function of the sequence. An alternative to be emphasized here is the power series whose reversion (or compositional inverse) is the generating function of the solution sequence.

The reasons for this choice are led by two: first, there seems to be a taxonomic advantage to cataloguing restricted dissection counts by the (easily obtained) closed forms of these power series; second, using the formulae of Lagrange and Newton we can often obtain summation formulae for nth terms of the solution sequence.

2. The Triangle-Free Case.

Let a_{n} denote the number of Schroeder dissections of the (n + 2)-gon having no triangular tiles. If we write A = A(x) = ∑_{k=0}^{∞} a_{n}x^{n}, then the tautological equation A = 1 + x^{2}A^{3} + x^{3}A^{4} + x^{4}A^{5} + ⋯ is easy to see ([S1]). Employing a formal geometric series and a change of variable to F = xA, we quickly obtain x = \frac{F −F^{2}−F^{3}}{1−F} := α(F). We call α the reversive symbol for the sequence \{a_{n}\}. According to the Lagrange Inversion Formula,

a_{n−1} = \frac{1}{n}([t^{n−1}](\frac{t}{α(t)})^{n}), n = 1, 2, …

and so Newton’s formula (’binomial series’) gives

a_{n} = \frac{1}{n + 1} ∑_{k=0}^{⌈ \frac{n−1}{2} ⌉} \binom{n + k}{k}\binom{n − k − 1}{k − 1}

where we agree to accept a_{0} = 1 in the case of the 2-gon.

Typeset by AMS-TEX 2 LEONARD M. SMILEY

1991 Mathematics Subject Classification. Primary 05A15 ; Secondary 11B83.

3. Odds and Evens.

If we restrict to Schroeder dissections all of whose tiles have an odd number of sides, the reversive symbol is α(F) = \frac{F −F^{2}−F^{3}}{1−F^{2}} , and

a_{n} = \frac{1}{n + 1} ∑_{k=0}^{⌈ \frac{n+1}{2} ⌉} \binom{2n − 2k}{n − 2k}\binom{n − k − 1}{k}

follows as before. Here we accept the anomaly that the 2-gon (n = 0) has −1 such dissections! For more detail on this example and the next, see [S2].

If all tiles are required to have an even number of sides, the reversive symbol is α(F) = \frac{F −2F^{3}}{1−F^{2}} . In this case, Newton’s formula insists that a_{2m+1} = 0, as we presumed, and

a_{2m} = \frac{1}{2m + 1} ∑_{k=0}^{m} \binom{2m + k}{k}\binom{m − 1}{k − 1}.

We note here that the sequence {a_{2m}} has been derived in a seemingly different context by Carlitz [C].

4. Conclusion The reversive symbol is useful for the taxonomy of many integer sequences. For the Schroeder numbers

themselves we have α(F) = \frac{F −2F^{2}}{1−F}, and

s_{n} = \frac{1}{n + 1} ∑_{k=0}^{n+1} \binom{2n − k}{n}\binom{n − 1}{k}

while the simplicity of the Catalan symbol α(F) = F − F^{2} produces the usual closed formula.

Of course, dissection enumeration is not the only application: the Motzkin numbers, M_{n}, which count the number of ways to place disjoint chords in a circle using n labelled points as endpoints, have symbol α(F) = \frac{F}{1+F +F^{2}} = \frac{F −F^{2}}{1−F^{3}} , yielding immediately

M_{n} = \frac{1}{n + 1} ∑_{k=0}^{n+1} \binom{n + 1}{k}\binom{k}{2k − n − 2}.

The author maintains a web-page catalogue of symbols and formulae in the same vein, to which visits and submissions are welcome [S3].

Acknowledgements

I wish to thank D. Zeilberger for the use of EKHAD, and for his E-mailed encouragement to publicize the new dissection counts. Thanks also to Neil Sloane for his interest.

References

[C] L. Carlitz, Enumeration of Two-line Arrays, Fibonacci Quarterly 11 (1973), 113-130.

[S1]  Smiley, L., A Nameless Number, http://saturn.math.uaa.alaska.edu/∼smiley/vsd1.html.
[S2]  Smiley, L., Dissections and Reversions, http://saturn.math.uaa.alaska.edu/∼smiley/vsd2.html.
[S3]  Smiley, L., Living Catalogue of Reversive Symbols, http://saturn.math.uaa.alaska.edu/∼smiley/vsd3.html.

Len Smiley, Dept. of Mathematical Sciences, UAA, Anchorage AK 99508

   E-mail address: smiley@math.uaa.alaska.edu