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A334257 Triangle read by rows: T(n,k) is the number of ordered pairs of n-permutations with exactly k common double descents, n>=0, 0<=k<=max{0,n-2}.

%I A334257 #29 Apr 28 2020 13:40:48
%S A334257 1,1,4,35,1,545,30,1,13250,1101,48,1,463899,51474,2956,70,1,22106253,
%T A334257 3070434,217271,7545,96,1,1375915620,229528818,19372881,864632,20322,
%U A334257 126,1,108386009099,21107789247,2070917370,113587335,3530099,61089,160,1
%N A334257 Triangle read by rows: T(n,k) is the number of ordered pairs of n-permutations with exactly k common double descents, n>=0, 0<=k<=max{0,n-2}.
%C A334257 An ordered pair of n-permutations ((a_1,a_2,...,a_n),(b_1,b_2,...,b_n)) has a common double descent at position i, 1<=i<=n-2, if a_i > a_i+1 > a_i+2 and b_i > b_i+1 > b_i+2.
%D A334257 R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.
%H A334257 Alois P. Heinz, <a href="/A334257/b334257.txt">Rows n = 0..60, flattened</a>
%H A334257 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 209.
%e A334257 T(4,1) = 30:  There are 9 such ordered pairs formed from the permutations 3421,2431,1432.  There are 9 such ordered pairs formed from the permutations 4312,4213,3214.  Then pairing each of these 6 permutations with 4321 gives 12 more ordered pairs with exactly 1 common double descent.  9+9+12 = 30.
%e A334257 Triangle T(n,k) begins:
%e A334257        1;
%e A334257        1;
%e A334257        4;
%e A334257       35,     1;
%e A334257      545,    30,    1;
%e A334257    13250,  1101,   48,  1;
%e A334257   463899, 51474, 2956, 70, 1;
%e A334257   ...
%p A334257 b:= proc(n, u, v, t) option remember; expand(`if`(n=0, 1,
%p A334257       add(add(b(n-1, u-j, v-i, x)*t, i=1..v)+
%p A334257           add(b(n-1, u-j, v+i-1, 1), i=1..n-v), j=1..u)+
%p A334257       add(add(b(n-1, u+j-1, v-i, 1), i=1..v)+
%p A334257           add(b(n-1, u+j-1, v+i-1, 1), i=1..n-v), j=1..n-u)))
%p A334257     end:
%p A334257 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2, 1)):
%p A334257 seq(T(n), n=0..10);  # _Alois P. Heinz_, Apr 26 2020
%t A334257 nn = 8; a = Apply[Plus,Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> y - 1; Map[Select[#, # > 0 &] &,
%t A334257   Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Grid
%Y A334257 Column k=0 gives A334412.
%Y A334257 Cf. A192721, A162975.
%K A334257 nonn,tabf
%O A334257 0,3
%A A334257 _Geoffrey Critzer_, Apr 26 2020