A362589 - OEIS

%I #75 May 02 2023 09:01:37

%S 1,1,4,1,36,18,1,576,432,68,1,14400,14400,3900,250,1,518400,648000,

%T 252000,32400,922,1,25401600,38102400,19404000,3880800,262542,3430,1,

%U 1625702400,2844979200,1795046400,493920000,56664384,2119152,12868,1

%N 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 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 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 Column k=1: Sum_{k=1..n-1} A192721(n,k)*k gives total number of common descents over all permutation pairs.

%e Triangle begins:

%e      1;

%e      1;

%e      4,     1;

%e     36,    18,    1;

%e    576,   432,   68,   1;

%e  14400, 14400, 3900, 250, 1;

%e  ...

%t 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 Cf. A001044 (column k=0), A102221 (row sums), A192721.

%K nonn,tabf

%O 0,3

%A _Geoffrey Critzer_, May 01 2023