A056972 Number of (binary) heaps on n levels (i.e., of 2^n - 1 elements).

1, 1, 2, 80, 21964800, 74836825861835980800000, 2606654998899867556195703676289609067340669424836280320000000000

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn

Comments

A sequence {a_i}{i=1..N} forms a (binary) heap if it satisfies a_i<a{2i} and a_i

a(n) is also the number of knockout tournament seedings that satisfy the increasing competitive intensity property. - Alexander Karpov, Aug 18 2015

a(n) is the number of coalescence sequences, or labeled histories, for a binary, leaf-labeled, rooted tree topology with 2^n leaves (Rosenberg 2006, Theorem 3.3 for a completely symmetric tree with 2^n leaves, dividing by Theorem 3.2 for 2^n leaves). - Noah A Rosenberg, Feb 12 2019

a(n+1) is also the number of random walk labelings of the perfect binary tree of height n, that begin at the root. - Sela Fried, Aug 02 2023

If a bracket of 2^n teams is specified for a single-elimination tournament, a(n) is the number of sequences in which the games of the tournament can be played in a single arena. - Noah A Rosenberg, Jan 01 2025

Examples

			There is 1 heap on 2^0-1=0 elements, 1 heap on 2^1-1=1 element and there are 2 heaps on 2^2-1=3 elements and so on.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..9
• Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307.
• Sela Fried and Toufik Mansour, Random walk labelings of perfect trees and other graphs, arXiv:2308.00315 [math.CO], 2023.
• Alexander Karpov, A theory of knockout tournament seedings, Heidelberg University, AWI Discussion Paper Series, No. 600.
• Matthew C. King and Noah A. Rosenberg, A mathematical connection between single-elimination sports tournaments and evolutionary trees, Math. Mag. 96 (2023), 484-497.
• Egor Lappo and Noah A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
• Noah A. Rosenberg, The Mean and Variance of the Numbers of r-Pronged Nodes and r-Caterpillars in Yule-Generated Genealogical Trees, Ann. Combinator. 10 (2006), 129-146.
• Eric Weisstein's World of Mathematics, Heap
• Wikipedia, Binary heap

Crossrefs

Cf. A000225, A056971, A195581.

Column k=2 of A273712.

Programs

Maple

a:= n-> (2^n-1)!/mul((2^k-1)^(2^(n-k)), k=1..n):
seq(a(i), i=0..6);  # Alois P. Heinz, Nov 22 2007

Mathematica

s[1] := 1; s[l_] := s[l] := Binomial[2^l-2, 2^(l-1)-1]s[l-1]^2; Table[s[l], {l, 10}]

Python

from math import prod, factorial
def A056972(n): return factorial((1<Chai Wah Wu, May 06 2024

Formula

a(n) = A056971(A000225(n)).

a(n) = A195581(2^n-1,n).

a(n) = binomial(2^n-2, 2^(n-1)-1)*a(n-1)^2. - Robert Israel, Aug 18 2015, from the Mathematica program

a(n) = (2^n-1)!/Product_{k=1..n} (2^k-1)^(2^(n-k)). - Robert Israel, Aug 18 2015, from the Maple program

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