A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280

Offset: 2

Author

Steven Finch, Nov 13 2021

Keywords

• nonn
• tabf

Comments

A round means the same as a directed ring or circle.

Examples

			Triangle starts:
[2]     2;
[3]     3;
[4]     8,     12;
[5]    30,     60;
[6]   144,    330,    120;
[7]   840,   2100,   1260;
[8]  5760,  15344,  11760,  1680;
[9] 45360, 127008, 113400, 30240;
...
For n = 4, there are 8 ways to make one round and 12 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 A066166 (Stanley's children's game).

Column 1 gives A001048.

Right border element of row n is A001813(n/2) = |A067994(n)| for even n.

Cf. A008279, A265609, A331327.

Programs

Maple

ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
T := (n, k) -> n!*coeff(xcoeff(n), t, k):
seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # Peter Luschny, Nov 13 2021
# second Maple program:
A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # Mélika Tebni, May 03 2023

Mathematica

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

Formula

G.f.: (1 - x)^(-x*t).

T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023

The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - Peter Luschny, May 03 2023