A393519 Triangular array: T(n,k) is the number of set partitions of [n] into k labeled 3-tori.

1, 0, 1, 0, 3, 1, 0, 6, 9, 1, 0, 36, 51, 18, 1, 0, 72, 360, 195, 30, 1, 0, 1080, 2412, 2025, 525, 45, 1, 0, 2160, 19656, 21042, 7875, 1155, 63, 1, 0, 50400, 177552, 225288, 113337, 23940, 2226, 84, 1, 0, 241920, 1557792, 2642112, 1614816, 456057, 61236, 3906, 108, 1

Offset

0,5

Comments

Definitions in link; 3-tori are the four space equivalent of cycles (two-space) and ordinary tori (rings, three-space).

References

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.

Links

Sean A. Irvine, Table of n, a(n) for n = 0..9869 (rows 0..139 flattened)

Marko Riedel, Set partitions into labeled 3-tori, definitions, derivation of recurrence, and a conjecture.

Formula

T(n,k) = n!*rho(n,k) where rho(n,k) = (1/n)*Sum_{q=0..n-1} rho(n-1-q,k-1) * tau3(q+1) with rho(0,k) = delta(k,0) and tau3(n) counts the number of ordered factorizations of n into three factors.

Example

n\k| 0   1    2    3   4  5
---+------------------------
  0| 1;
  1| 0,  1;
  2| 0,  3,   1;
  3| 0,  6,   9,   1;
  4| 0, 36,  51,  18,  1;
  5| 0, 72, 360, 195, 30, 1;

Crossrefs

Cf. A007425 (tau3), A338805 (three-space tori).

Row sums are A318695.

Sequence in context: A248826 A058152 A394438 * A058140 A074678 A363976

Adjacent sequences: A393516 A393517 A393518 * A393520 A393521 A393522

Keyword

nonn,tabl

Author

Marko Riedel, Feb 18 2026

Status

approved