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A321059 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=3.

%I A321059 #9 Oct 27 2018 12:01:17
%S A321059 2,8,64,852,17008,467696,16860608,769534688,43316046592,2945846638464,
%T A321059 238048843712512,22542120635302912,2472277850712350720,
%U A321059 310880199760083564032,44429722692959679590400,7161345941202962487965696,1292988538929694918729072640
%N A321059 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=3.
%H A321059 Alois P. Heinz, <a href="/A321059/b321059.txt">Table of n, a(n) for n = 3..110</a>
%e A321059 a(3) = 2: (312,312), (321,321).
%e A321059 a(4) = 8: (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421).
%p A321059 b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
%p A321059       add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
%p A321059       add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
%p A321059     end:
%p A321059 a:= n-> (k-> b(k-1, n-k, n-k))(3):
%p A321059 seq(a(n), n=3..20);
%Y A321059 Column k=3 of A262372.
%K A321059 nonn
%O A321059 3,1
%A A321059 _Alois P. Heinz_, Oct 26 2018