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A334394 Triangle read by rows: T(n,k) is the number of ordered triples of n-permutations with exactly k common descents, n>=0, 0<=k<=max(0,n-1).

%I A334394 #26 Apr 28 2020 15:39:56
%S A334394 1,1,7,1,163,52,1,8983,4499,341,1,966751,660746,98256,2246,1,
%T A334394 179781181,155729277,35677082,2045282,15177,1,53090086057,55690144728,
%U A334394 17446464519,1754605504,42658239,104952,1,23402291822743,28825420903351,11518335730323,1717307782339,84058424389,905365701,739153,1
%N A334394 Triangle read by rows: T(n,k) is the number of ordered triples of n-permutations with exactly k common descents, n>=0, 0<=k<=max(0,n-1).
%C A334394 An ordered triple of n-permutations ( (a_1,a_2,...,a_n),(b_1,b_2,...,b_n),(c_1,c_2,...,c_n) ) has a common descent at position i, 1<=i<=n-1, if a_i > a_i+1, b_i > b_i+1 and c_i > c_i+1.
%D A334394 R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.
%H A334394 Alois P. Heinz, <a href="/A334394/b334394.txt">Rows n = 0..30, flattened</a>
%H A334394 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 209.
%F A334394 Sum_{n>=0} Sum_{k>=0} T(n,k)*y^k*x^n/n!^3 = (y-1)/(y-f(x*(y-1))) where f(z) = Sum_{n>=0} z^n/n!^3.
%e A334394 Triangle begins:
%e A334394        1;
%e A334394        1;
%e A334394        7,      1;
%e A334394      163,     52,     1;
%e A334394     8983,   4499,   341,   1;
%e A334394   966751, 660746, 98256, 2246, 1;
%e A334394   ...
%p A334394 T:= (n, k)-> n!^3*coeff(series(coeff(series((y-1)/(y-add((x*
%p A334394     (y-1))^j/j!^3, j=0..n)), y, k+1), y, k), x, n+1), x, n):
%p A334394 seq(seq(T(n,k), k=0..max(0, n-1)), n=0..10);  # _Alois P. Heinz_, Apr 28 2020
%t A334394 nn = 6; e3[x_] := Sum[x^n/n!^3, {n, 0, nn}];Drop[Map[Select[#, # > 0 &] &,
%t A334394    Table[n!^3, {n, 0, nn}] CoefficientList[Series[(y - 1)/(y - e3[x (y - 1)]), {x, 0, nn}], {x, y}]], 1] // Grid
%Y A334394 Cf. A192721, A008292, A212856 (column k=0), A000442 (row sums).
%K A334394 nonn,tabf
%O A334394 0,3
%A A334394 _Geoffrey Critzer_, Apr 26 2020