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A271023 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i such that i and j are in the same block; triangle T(n,k), n>=0, 0<=k<=n*(n-1)/2 read by rows.

%I A271023 #26 Apr 29 2022 12:11:24
%S A271023 1,1,1,1,1,3,0,1,1,6,3,4,0,0,1,1,10,15,10,10,0,5,0,0,0,1,1,15,45,35,
%T A271023 60,0,25,15,0,0,6,0,0,0,0,1,1,21,105,140,210,105,105,105,0,35,21,21,0,
%U A271023 0,0,7,0,0,0,0,0,1,1,28,210,476,665,840,350,700,210
%N A271023 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i<j such that i and j are in the same block; triangle T(n,k), n>=0, 0<=k<=n*(n-1)/2 read by rows.
%H A271023 Alois P. Heinz, <a href="/A271023/b271023.txt">Rows n = 0..40, flattened</a>
%F A271023 T(n,k) = A271024(n,n*(n-1)/2-k).
%F A271023 Sum_{k=0..n*(n-1)/2} k * T(n,k) = A105488(n+2) for n > 1.
%e A271023 T(3,0) = 1: 1|2|3.
%e A271023 T(3,1) = 3: 12|3, 13|2, 1|23.
%e A271023 T(3,3) = 1: 123.
%e A271023 Triangle T(n,k) begins:
%e A271023   1;
%e A271023   1;
%e A271023   1,  1;
%e A271023   1,  3,  0,  1;
%e A271023   1,  6,  3,  4,  0, 0,  1;
%e A271023   1, 10, 15, 10, 10, 0,  5,  0, 0, 0, 1;
%e A271023   1, 15, 45, 35, 60, 0, 25, 15, 0, 0, 6, 0, 0, 0, 0, 1;
%p A271023 b:= proc(n, l) option remember; `if`(n=0, x^
%p A271023       add(j*(j-1)/2, j=l), b(n-1, [l[], 1])+
%p A271023       add(b(n-1, subsop(j=l[j]+1, l)), j=1..nops(l)))
%p A271023     end:
%p A271023 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [])):
%p A271023 seq(T(n), n=0..10);
%t A271023 b[n_, l_] := b[n, l] = If[n == 0, x^Sum[j*(j-1)/2, {j, l}], b[n-1, Append[l, 1]] + Sum[b[n-1, ReplacePart[l, j -> l[[j]]+1]], {j, 1, Length[l]}]];
%t A271023 T[n_] := CoefficientList[b[n, {}], x];
%t A271023 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 29 2022, after _Alois P. Heinz_ *)
%Y A271023 Columns k=0-2 give: A000012, A161680, A050534(n-1) for n>0.
%Y A271023 Row sums give A000110.
%Y A271023 Cf. A105488, A161680, A271024.
%K A271023 nonn,look,tabf
%O A271023 0,6
%A A271023 _Alois P. Heinz_, Mar 28 2016