A244108 Number T(n,k) of permutations of {1,2,...,n} that result in a binary search tree of height k; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

1, 1, 2, 2, 4, 16, 40, 80, 80, 8, 64, 400, 2240, 11360, 55040, 253440, 1056000, 3801600, 10982400, 21964800, 21964800, 16, 208, 2048, 18816, 168768, 1508032, 13501312, 121362560, 1099169280, 10049994240, 92644597760, 857213660160, 7907423180800, 72155129446400

Offset: 0

Author

Alois P. Heinz, Dec 21 2015

Keywords

• nonn
• tabf

Comments

Empty external nodes are counted in determining the height of a search tree.

Examples

			Triangle T(n,k) begins:
: 1;
:    1;
:       2;
:       2,  4;
:          16,      8;
:          40,     64,      16;
:          80,    400,     208,      32;
:          80,   2240,    2048,     608,     64;
:               11360,   18816,    8352,   1664,   128;
:               55040,  168768,  104448,  30016,  4352,   256;
:              253440, 1508032, 1277568, 479040, 99200, 11008, 512;
		

Links

• Alois P. Heinz, Columns k = 0..9, flattened
• Wikipedia, Binary search tree
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Row sums give A000142.

Column sums give A227822.

Main diagonal gives A011782, lower diagonal gives A076616.

T(n,A000523(n)+1) = A076615(n).

T(2^n-1,n) = A056972(n).

T(2n,n) = A265846(n).

Cf. A195581 (the same read by rows), A195582, A195583, A316944, A317012.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, `if`(k b(n, k)-b(n, k-1):
seq(seq(T(n, k), n=k..2^k-1), k=0..5);

Mathematica

b[n_, k_] := b[n, k] = If[n<2, If[kJean-François Alcover, Feb 19 2017, translated from Maple *)

Formula

Sum_{k=0..n} k * T(n,k) = A316944(n).

Sum_{k=n..2^n-1} k * T(k,n) = A317012(n).