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A202363 Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.

%I A202363 #22 Nov 22 2023 15:38:18
%S A202363 1,6,3,36,24,12,240,180,120,60,1800,1440,1080,720,360,15120,12600,
%T A202363 10080,7560,5040,2520,141120,120960,100800,80640,60480,40320,20160,
%U A202363 1451520,1270080,1088640,907200,725760,544320,362880,181440,16329600,14515200,12700800,10886400,9072000,7257600,5443200,3628800,1814400
%N A202363 Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.
%C A202363 Row sums = A001809.
%C A202363 Column for k = 0 is A001286.
%H A202363 Alois P. Heinz, <a href="/A202363/b202363.txt">Rows n = 2..142, flattened</a>
%F A202363 E.g.f.: x^2/2 * (1/(1-x)^2)* (1/(1-y*x)).
%e A202363 T(3,1) = 3 because from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have respectively 3 inversion pairs (1,2), (2,3) and (1,3) which are all separated by 1 element.
%e A202363 Triangle T(n,k) begins:
%e A202363        1;
%e A202363        6,      3;
%e A202363       36,     24,     12;
%e A202363      240,    180,    120,    60;
%e A202363     1800,   1440,   1080,   720,   360;
%e A202363    15120,  12600,  10080,  7560,  5040,  2520;
%e A202363   141120, 120960, 100800, 80640, 60480, 40320, 20160;
%e A202363   ...
%t A202363 nn=10;Range[0,nn]!CoefficientList[Series[x^2/2/(1-x)^2/(1-y x),{x,0,nn}],{x,y}]//Grid
%Y A202363 Cf. A001286, A001809, A055303.
%K A202363 nonn,tabl
%O A202363 2,2
%A A202363 _Geoffrey Critzer_, Jan 09 2013