A076616 Number of permutations of {1,2,...,n} that result in a binary search tree (when elements of the permutation are inserted in that order) of height n-1 (i.e., the second largest possible height).
0, 0, 0, 2, 16, 64, 208, 608, 1664, 4352, 11008, 27136, 65536, 155648, 364544, 843776, 1933312, 4390912, 9895936, 22151168, 49283072, 109051904, 240123904, 526385152, 1149239296, 2499805184, 5419040768, 11710496768, 25232932864, 54223962112, 116232552448
Offset: 0
Examples
a(3) = 2 because only the permutations (2,1,3) and (2,3,1) result in a search tree of height 2 (notice we count empty external nodes in determining the height). The largest such trees are of height 3.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
Programs
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Maple
a:= n-> max(-(<<0|1|0>, <0|0|1>, <8|-12|6>>^n. <<1/2, 1, 1>>)[1$2], 0): seq(a(n), n=0..40); # Alois P. Heinz, Sep 20 2011
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Mathematica
Join[{0, 0, 0}, LinearRecurrence[{6, -12, 8}, {2, 16, 64}, 40]] (* Jean-François Alcover, Jan 09 2025 *) -
PARI
concat(vector(3), Vec(2*x^3*(1+2*x-4*x^2)/(1-2*x)^3 + O(x^50))) \\ Colin Barker, May 16 2016
Formula
G.f.: 2*(4*x^2-2*x-1)*x^3/(2*x-1)^3. - Alois P. Heinz, Sep 20 2011
From Colin Barker, May 16 2016: (Start)
a(n) = 2^(n-3)*(n^2-n-4) for n>2.
a(n) = 6*a(n-1)-12*a(n-2)+8*a(n-3) for n>5.
(End)
From Alois P. Heinz, May 31 2022: (Start)
a(n) = 2 * A100312(n-3) for n>=3.
Extensions
More terms from Alois P. Heinz, Sep 20 2011