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A228924 Irregular triangular array read by rows: T(n,k) is the number of derangement permutations of [n] that have exactly k inversions; n>=2, 1<=k<=binomial(n,2) for even n, 1<=k<=binomial(n,2)-1 for odd n.

%I A228924 #26 Mar 06 2025 18:15:31
%S A228924 1,0,2,0,1,4,1,2,1,0,0,4,8,4,10,10,6,2,0,0,1,12,18,16,35,44,47,40,25,
%T A228924 14,8,4,1,0,0,0,6,32,44,60,118,160,208,244,244,214,174,140,104,64,30,
%U A228924 10,2,0,0,0,1,24,83,118,206,388,565,802,1068,1308,1466,1508,1479,1414,1290,1076,806,544,333,186,96,46,19,6,1
%N A228924 Irregular triangular array read by rows: T(n,k) is the number of derangement permutations of [n] that have exactly k inversions; n>=2, 1<=k<=binomial(n,2) for even n, 1<=k<=binomial(n,2)-1 for odd n.
%C A228924 Row sums = A000166.
%C A228924 Sum_{k>=1} T(n,k)*k = A216239(n).
%C A228924 Sum_{even k} T(n,k) = A003221(n) and Sum_{odd k} T(n,k) = A000387(n).
%C A228924 It would be nice to have a closed formula for T(n,k). - _Alois P. Heinz_, Dec 31 2014
%H A228924 Alois P. Heinz, <a href="/A228924/b228924.txt">Rows n = 2..23, flattened</a>
%e A228924 Triangle T(n,k) begins:
%e A228924   1;
%e A228924   0, 2;
%e A228924   0, 1, 4,  1,  2,  1;
%e A228924   0, 0, 4,  8,  4, 10, 10,  6,  2;
%e A228924   0, 0, 1, 12, 18, 16, 35, 44, 47, 40, 25, 14, 8, 4, 1;
%e A228924   ...
%t A228924 Map[Distribution[#,Range[1,Max[#]]]&,Table[Map[Inversions,Derangements[n]],{n,2,6}]]//Grid
%Y A228924 Cf. A000166, A000387, A003221, A008302, A216239.
%K A228924 nonn,tabf
%O A228924 2,3
%A A228924 _Geoffrey Critzer_, Sep 08 2013