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}.

1, 1, 4, 1, 36, 18, 1, 576, 432, 68, 1, 14400, 14400, 3900, 250, 1, 518400, 648000, 252000, 32400, 922, 1, 25401600, 38102400, 19404000, 3880800, 262542, 3430, 1, 1625702400, 2844979200, 1795046400, 493920000, 56664384, 2119152, 12868, 1

Offset

0,3

Links

Table of n, a(n) for n=0..36.

L. Carlitz, R. Scoville and T. Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc., 80 (1974), 881-884.

Formula

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.

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

Example

Triangle begins:
     1;
     1;
     4,     1;
    36,    18,    1;
   576,   432,   68,   1;
 14400, 14400, 3900, 250, 1;
 ...

Mathematica

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

Crossrefs

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

Sequence in context: A144267 A011801 A169656 * A303987 A297900 A363819

Adjacent sequences: A362586 A362587 A362588 * A362590 A362591 A362592

Keyword

nonn,tabf

Author

Geoffrey Critzer, May 01 2023

Status

approved