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A362589 Triangular array read by rows. T(n,k) is the number of ways to form an ordered pair of n-permutations and then choose a size k subset of its common descent set, n >= 0, 0 <= k <= max{0,n-1}.

%I A362589 #75 May 02 2023 09:01:37
%S A362589 1,1,4,1,36,18,1,576,432,68,1,14400,14400,3900,250,1,518400,648000,
%T A362589 252000,32400,922,1,25401600,38102400,19404000,3880800,262542,3430,1,
%U A362589 1625702400,2844979200,1795046400,493920000,56664384,2119152,12868,1
%N A362589 Triangular array read by rows.  T(n,k) is the number of ways to form an ordered pair of n-permutations and then choose a size k subset of its common descent set, n >= 0, 0 <= k <= max{0,n-1}.
%H A362589 L. Carlitz, R. Scoville and T. Vaughan, <a href="https://doi.org/10.1090/S0002-9904-1974-13554-8">Enumeration of pairs of permutations and sequences</a>, Bull. Amer. Math. Soc., 80 (1974), 881-884.
%F A362589 Sum_{n>=0} Sum_{k=0..n-1} T(n,k)*u^k*z^n/(n!)^2 = u/(u + 1 - E(u*z)) where E(z) = Sum_{n>=0} z^n/(n!)^2.
%F A362589 Column k=1: Sum_{k=1..n-1} A192721(n,k)*k gives total number of common descents over all permutation pairs.
%e A362589 Triangle begins:
%e A362589      1;
%e A362589      1;
%e A362589      4,     1;
%e A362589     36,    18,    1;
%e A362589    576,   432,   68,   1;
%e A362589  14400, 14400, 3900, 250, 1;
%e A362589  ...
%t A362589 nn = 8; B[n_] := n!^2; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[u/(u + 1 - e[u z]), {z, 0, nn}], {z, u}]] // Flatten
%Y A362589 Cf. A001044 (column k=0), A102221 (row sums), A192721.
%K A362589 nonn,tabf
%O A362589 0,3
%A A362589 _Geoffrey Critzer_, May 01 2023