A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

3, 8, 30, 144, 90, 840, 840, 5760, 7280, 45360, 66528, 7560, 403200, 657720, 151200, 3991680, 7064640, 2356200, 43545600, 82285632, 34890240, 1247400, 518918400, 1035365760, 521080560, 43243200, 6706022400, 14013679680, 8034586560, 1059458400

Offset: 3

Author

Steven Finch, Nov 17 2021

Keywords

• nonn
• tabf

Comments

A nontrivial round means the same as a ring or circle consisting of more than one child.

Examples

			Triangle starts:
[3]           3;
[4]           8;
[5]          30;
[6]         144,          90;
[7]         840,         840;
[8]        5760,        7280;
[9]       45360,       66528,       7560;
[10]     403200,      657720,     151200;
[11]    3991680,     7064640,    2356200;
[12]   43545600,    82285632,   34890240,    1247400;
[13]  518918400,  1035365760,  521080560,   43243200;
[14] 6706022400, 14013679680, 8034586560, 1059458400;
...
For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.
		

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

• Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Crossrefs

Row sums give A066165 (variant of Stanley's children's game).

Column 1 gives A001048.

Right border element of row n is A166334(n/3) for each n divisible by 3.

Cf. A066166, A349280 (correspond to Stanley's original game).

Programs

Mathematica

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]

Formula

E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).