A370237 Number of genus 3 partitions of the n-set.

1, 94, 2620, 45430, 600655, 6633484, 64336844, 565256120

Offset: 8

Author

Robert Coquereaux, Feb 12 2024

Keywords

• nonn
• more

Comments

Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 3) with B(8, 3) = 1.

a(8) = 1 through a(15) = 565256120 were explicitly determined by listing of partitions of an n-set and selecting those of genus 3.

The coefficients of the sixth-degree polynomial appearing in the numerator of the conjectured formula were determined by using experimental values for a(8) up to a(14); the term a(15) given by the formula agrees with the experimental value.

Using the conjectured formula for a(n) gives the following terms for n=16..20 : 4593034160, 35025118700, 253374008888, 1753071498620, 11675101781850. The E.g.f. given in the Formula section is obtained from the conjectured formula for a(n).

Links

• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.
• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 9. See also arXiv:2305.01100, 2023.

Crossrefs

Column g=3 of A370235.

Cf. A000108, A002802, A297179, A001263, A370236, A297178.

Formula

Conjecture: a(n) = (1/(2^13 * 3^4 * 5 * 7)) * (35*n^6 - 819*n^5 + 7589*n^4 - 36009*n^3 + 93464*n^2 - 129060*n + 95040)/((2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)) * (1/(n-8)!) * (2*n)!/n!.

Conjecture: E.g.f.: (1/181440)*exp(2*x)*(x^2*(720 - 720*x + 1080*x^2 - 720*x^3 + 537*x^4 - 294*x^5 + 140*x^6)*BesselI(0, 2*x) + x*(-720 + 720*x - 1440*x^2 + 1080*x^3 - 1017*x^4 + 594*x^5 - 329*x^6 + 140*x^7)*BesselI(1, 2*x)).