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A349280
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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).
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2
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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
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OFFSET
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2,1
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COMMENTS
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A round means the same as a directed ring or circle.
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
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LINKS
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FORMULA
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G.f.: (1 - x)^(-x*t).
T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023
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EXAMPLE
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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.
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MAPLE
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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)):
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MATHEMATICA
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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]}]
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CROSSREFS
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Row sums give A066166 (Stanley's children's game).
Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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