Riordan number

A Riordan number is a natural number of set partitions of n points placed on a circle such that every set block contains at least two points and the convex hulls of distinct blocks are disjoint. They are denoted as Rn. These was first documented by Frank R. Bernhart in 1999, who named it after mathematician John Riordan. It appears often in enumeration problems. It is related to Catalan numbers and Motzkin numbers.

Riordan numbers are also known as Motzkin summands or ring numbers.

The first few Riordan numbers are:

1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595, 13393689, ... (sequence A005043 in the OEIS)

Generating function

The generating function for Riordan numbers is:

R (x) =∑n=0∞Rn xn
=12x(1−1−3x1+x)
=1+x2+x3+3 x4+6 x5+15 x6+...
=1+x−1−2x−3x22x(1+x)

which follow the functional equation

R (x)=11+x+x R (x)2.

References

  1. 1 2 3 Bernhart, Frank R. (June 1999). "Catalan, Motzkin, and Riordan numbers". Discrete Mathematics. 204 (1–3): 73–112. doi:10.1016/S0012-365X(99)00054-0.
  2. 1 2 3 4 Weisstein, Eric W. "Riordan Number". Wolfram MathWorld. Wolfram Research, Inc. Retrieved 2026-11-11.
  3. ↑ Sloane, N. J. A. (ed.). "Sequence A005043 (Riordan numbers: a(n) = (n-1)*(2*a(n-1) + 3*a(n-2))/(n+1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2026-11-11.
  4. ↑ Aigner, Martin (June 2008). "Enumeration via ballot numbers". Discrete Mathematics. 308 (12): 2544–2563. doi:10.1016/j.disc.2007.06.012.
  5. ↑ Barry, Paul (2018-02-09). "On a transformation of Riordan moment sequences". arXiv:1802.03443 [math.CO].
  6. ↑ Braha, Naim L.; Mansour, Toufik (March 2025). "Some properties of new sequence spaces based on Riordan numbers". Journal of Mathematical Analysis and Applications. 543 (2): 128902. doi:10.1016/j.jmaa.2024.128902.{{cite journal}}: CS1 maint: article number as page number (link)