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A262372 Number T(n,k) of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=k if n>0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A262372 #23 Apr 14 2025 15:14:20
%S A262372 1,0,1,0,1,1,0,2,2,2,0,10,8,8,10,0,88,68,64,68,88,0,1216,952,852,852,
%T A262372 952,1216,0,24176,19312,17008,16328,17008,19312,24176,0,654424,533544,
%U A262372 467696,438496,438496,467696,533544,654424
%N A262372 Number T(n,k) of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=k if n>0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A262372 Alois P. Heinz, <a href="/A262372/b262372.txt">Rows n = 0..100, flattened</a>
%e A262372 T(4,1) = 10: (1234,1234), (1243,1243), (1243,1342), (1324,1324), (1324,1423), (1342,1243), (1342,1342), (1423,1324), (1423,1423), (1432,1432).
%e A262372 T(4,2) = 8: (2134,2134), (2143,2143), (2314,2314), (2314,2413), (2341,2341), (2413,2314), (2413,2413), (2431,2431).
%e A262372 T(4,3) = 8: (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421).
%e A262372 T(4,4) = 10: (4123,4123), (4132,4132), (4132,4231), (4213,4213), (4213,4312), (4231,4132), (4231,4231), (4312,4213), (4312,4312), (4321,4321).
%e A262372 Triangle T(n,k) begins:
%e A262372   1
%e A262372   0,     1;
%e A262372   0,     1,     1;
%e A262372   0,     2,     2,     2;
%e A262372   0,    10,     8,     8,    10;
%e A262372   0,    88,    68,    64,    68,    88;
%e A262372   0,  1216,   952,   852,   852,   952,  1216;
%e A262372   0, 24176, 19312, 17008, 16328, 17008, 19312, 24176;
%e A262372   ...
%p A262372 b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
%p A262372       add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
%p A262372       add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
%p A262372     end:
%p A262372 T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(k-1, n-k, n-k)):
%p A262372 seq(seq(T(n, k), k=0..n), n=0..10);
%t A262372 b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1,
%t A262372   Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}, {j, 1, u}] +
%t A262372   Sum[b[u + j - 1, o - j, h - i], {i, 1, h}, {j, 1, o}]];
%t A262372 T[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[k - 1, n - k, n - k]];
%t A262372 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 05 2019, after _Alois P. Heinz_ *)
%Y A262372 Main diagonal and column k=1 give A060350(n-1) for n>0.
%Y A262372 Columns k=0,2-10 give: A000007, A262479, A321059, A321060, A321061, A321062, A321063, A321064, A321065, A321066.
%Y A262372 Row sums give A262234.
%Y A262372 T(2n,n) gives A262379.
%K A262372 nonn,look,tabl
%O A262372 0,8
%A A262372 _Alois P. Heinz_, Sep 20 2015