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.

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, 952, 1216, 0, 24176, 19312, 17008, 16328, 17008, 19312, 24176, 0, 654424, 533544, 467696, 438496, 438496, 467696, 533544, 654424

Offset: 0

Author

Alois P. Heinz, Sep 20 2015

Keywords

• nonn
• look
• tabl

Examples

			T(4,1) = 10: (1234,1234), (1243,1243), (1243,1342), (1324,1324), (1324,1423), (1342,1243), (1342,1342), (1423,1324), (1423,1423), (1432,1432).
T(4,2) = 8: (2134,2134), (2143,2143), (2314,2314), (2314,2413), (2341,2341), (2413,2314), (2413,2413), (2431,2431).
T(4,3) = 8: (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421).
T(4,4) = 10: (4123,4123), (4132,4132), (4132,4231), (4213,4213), (4213,4312), (4231,4132), (4231,4231), (4312,4213), (4312,4312), (4321,4321).
Triangle T(n,k) begins:
  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,   952,  1216;
  0, 24176, 19312, 17008, 16328, 17008, 19312, 24176;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened

Crossrefs

Main diagonal and column k=1 give A060350(n-1) for n>0.

Columns k=0,2-10 give: A000007, A262479, A321059, A321060, A321061, A321062, A321063, A321064, A321065, A321066.

Row sums give A262234.

T(2n,n) gives A262379.

Programs

Maple

b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(k-1, n-k, n-k)):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1,
  Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}, {j, 1, u}] +
  Sum[b[u + j - 1, o - j, h - i], {i, 1, h}, {j, 1, o}]];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[k - 1, n - k, n - k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, after Alois P. Heinz *)