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A306191 T(n,k) is a triangular array read by rows. Let S_n act on the set of size two subsets of {1,2,...,n}. T(n,k) is the number of permutations in S_n that fix exactly k size two subsets, n >= 1, 0 <= k <= binomial(n,2).

%I A306191 #11 Feb 17 2019 09:29:22
%S A306191 1,0,2,2,3,0,1,14,0,9,0,0,0,1,54,40,15,0,10,0,0,0,0,0,1,304,300,0,100,
%T A306191 0,0,0,15,0,0,0,0,0,0,0,1,2260,1638,630,315,0,105,70,0,0,0,0,21,0,0,0,
%U A306191 0,0,0,0,0,0,1,18108,12992,5460,1344,1645,0,420,0,210,0,112,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,1
%N A306191 T(n,k) is a triangular array read by rows.  Let S_n act on the set of size two subsets of {1,2,...,n}.  T(n,k) is the number of permutations in S_n that fix exactly k size two subsets, n >= 1, 0 <= k <= binomial(n,2).
%C A306191 The action of S_n on the 2-subsets of {1,2,...,n} is defined:  For all pi in S_n, pi({i,j}) = {pi(i),pi(j)}.
%e A306191 1,
%e A306191 0,   2,
%e A306191 2,   3,   0,  1,
%e A306191 14,  0,   9,  0,   0,  0, 1,
%e A306191 54,  40,  15, 0,   10, 0, 0, 0,  0, 0, 1,
%e A306191 304, 300, 0,  100, 0,  0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 1,
%t A306191 f[list_] := Flatten[Position[list /. x_ /; x > 0 -> 1, 1]];
%t A306191 Level[CoefficientList[Table[n! PairGroupIndex[SymmetricGroup[n], s] /. {Table[s[i] -> 1, {i, 2, Binomial[n, 2]}]}, {n, 1, 8}],
%t A306191    s[1]], {2}] // Grid
%Y A306191 Cf. A137482 is column 1.
%K A306191 nonn,tabf
%O A306191 1,3
%A A306191 _Geoffrey Critzer_, Jan 28 2019