A189232 - OEIS

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A189232

Triangle read by rows: Number of crossing set partitions of {1,2,...,n} into k blocks.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 5, 0, 0, 0, 16, 40, 15, 0, 0, 0, 42, 196, 175, 35, 0, 0, 0, 99, 770, 1211, 560, 70, 0, 0, 0, 219, 2689, 6594, 5187, 1470, 126, 0, 0, 0, 466, 8790, 31585, 37233, 17535, 3360, 210, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,12

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999 (Exericses 6.19)

LINKS

Table of n, a(n) for n=1..55.

FORMULA

T(n,k) = S2(n,k) - C(n,k-1)*C(n,k)/n; S2(n,k) Stirling numbers of the second kind, C(n,k) binomial coefficients.

EXAMPLE

There are 10 crossing set partitions of {1,2,3,4,5}.

T(5,2) = card{13|245, 14|235, 24|135, 25|134, 35|124} = 5.

T(5,3) = card{1|35|24, 2|14|35, 3|14|25, 4|13|25, 5|13|24} = 5.

[1] 0

[2] 0, 0

[3] 0, 0, 0

[4] 0, 1, 0, 0

[5] 0, 5, 5, 0, 0

[6] 0, 16, 40, 15, 0, 0

[7] 0, 42, 196, 175, 35, 0, 0

[8] 0, 99, 770, 1211, 560, 70, 0, 0

MAPLE

A189232 := (n, k) -> combinat[stirling2](n, k) - binomial(n, k-1)*binomial(n, k)/n:

for n from 1 to 9 do seq(A189232(n, k), k = 1..n) od;

MATHEMATICA

T[n_, k_] := StirlingS2[n, k] - Binomial[n, k-1] Binomial[n, k]/n;

Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 24 2019 *)

CROSSREFS

Row sums A016098, A001263.

Sequence in context: A371264 A065937 A197738 * A247667 A115144 A200506

Adjacent sequences: A189229 A189230 A189231 * A189233 A189234 A189235

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Apr 28 2011

STATUS

approved