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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.
3
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, 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, 0, 0, 7, 0, 0, 0, 0, 0, 1, 1, 28, 210, 476, 665, 840, 350, 700, 210
OFFSET
0,6
LINKS
FORMULA
T(n,k) = A271024(n,n*(n-1)/2-k).
Sum_{k=0..n*(n-1)/2} k * T(n,k) = A105488(n+2) for n > 1.
EXAMPLE
T(3,0) = 1: 1|2|3.
T(3,1) = 3: 12|3, 13|2, 1|23.
T(3,3) = 1: 123.
Triangle T(n,k) begins:
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, 60, 0, 25, 15, 0, 0, 6, 0, 0, 0, 0, 1;
MAPLE
b:= proc(n, l) option remember; `if`(n=0, x^
add(j*(j-1)/2, j=l), b(n-1, [l[], 1])+
add(b(n-1, subsop(j=l[j]+1, l)), j=1..nops(l)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [])):
seq(T(n), n=0..10);
MATHEMATICA
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[n_] := CoefficientList[b[n, {}], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)
CROSSREFS
Columns k=0-2 give: A000012, A161680, A050534(n-1) for n>0.
Row sums give A000110.
Sequence in context: A011256 A294212 A220691 * A370041 A143624 A126308
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Mar 28 2016
STATUS
approved