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A189232 Triangle read by rows: Number of crossing set partitions of {1,2,...,n} into k blocks.

%I A189232 #10 Jun 24 2019 03:38:16
%S A189232 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,
%T A189232 99,770,1211,560,70,0,0,0,219,2689,6594,5187,1470,126,0,0,0,466,8790,
%U A189232 31585,37233,17535,3360,210,0,0
%N A189232 Triangle read by rows: Number of crossing set partitions of {1,2,...,n} into k blocks.
%D A189232 R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999 (Exericses 6.19)
%F A189232 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.
%e A189232 There are 10 crossing set partitions of {1,2,3,4,5}.
%e A189232 T(5,2) = card{13|245, 14|235, 24|135, 25|134, 35|124} = 5.
%e A189232 T(5,3) = card{1|35|24, 2|14|35, 3|14|25, 4|13|25, 5|13|24} = 5.
%e A189232 [1] 0
%e A189232 [2] 0, 0
%e A189232 [3] 0, 0, 0
%e A189232 [4] 0, 1, 0, 0
%e A189232 [5] 0, 5, 5, 0, 0
%e A189232 [6] 0, 16, 40, 15, 0, 0
%e A189232 [7] 0, 42, 196, 175, 35, 0, 0
%e A189232 [8] 0, 99, 770, 1211, 560, 70, 0, 0
%p A189232 A189232 := (n,k) -> combinat[stirling2](n,k) - binomial(n,k-1)*binomial(n,k)/n:
%p A189232 for n from 1 to 9 do seq(A189232(n,k), k = 1..n) od;
%t A189232 T[n_, k_] := StirlingS2[n, k] - Binomial[n, k-1] Binomial[n, k]/n;
%t A189232 Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* _Jean-François Alcover_, Jun 24 2019 *)
%Y A189232 Row sums A016098, A001263.
%K A189232 nonn,tabl
%O A189232 1,12
%A A189232 _Peter Luschny_, Apr 28 2011