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

0,8

Links

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

Example

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;
  ...

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 *)

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.

Sequence in context: A307520 A265648 A181230 * A390550 A390108 A392180

Adjacent sequences: A262369 A262370 A262371 * A262373 A262374 A262375

Keyword

nonn,look,tabl

Author

Alois P. Heinz, Sep 20 2015

Status

approved