OFFSET
2,2
COMMENTS
Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T231 illustrated below corresponds to a treeshelf constructed from permutation 231. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.
a(n) is also the sum of the last entries in all blocks of all set partitions of [n-1]. a(4) = 23 because the sum of the last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+5+5+4+6 = 23. - Alois P. Heinz, Apr 24 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 2..575
Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), pp. 35-50.
FORMULA
E.g.f.: (z*e^z - e^z + 1)*e^(e^z-1).
a(n) = (n + 1)*b(n) - b(n+1) where b(n) is the n-th Bell number (see A000110).
Asymptotics: a(n) ~ n*b(n).
a(n) = Sum_{k=1..n-1} A285595(n-1,k)/k. - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=1..n} Stirling2(n,k) * (n-k). - Ilya Gutkovskiy, Apr 06 2021
a(n) ~ n*Bell(n)*(1 - 1/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=n-1..(n-1)*n/2} k * A367955(n-1,k). - Alois P. Heinz, Dec 11 2023
EXAMPLE
Treeshelves of size 3:
1 1 1 1 1 1
/ \ / \ / \ / \
2 2 / \ 2 \ / 2
/ \ 2 2 3 3
3 3 \ /
3 3
Pattern T231:
1
/
/
2
\
3
Treeshelves of size 3 that avoid pattern T231:
1 1 1 1 1
/ \ \ / \ / \
2 2 \ 2 \ / 2
/ \ 2 3 3
3 3 /
3
Popularity of left children here is 5.
MAPLE
b:= proc(n, m) option remember; `if`(n=0, [1, 0],
(p-> p+[0, p[1]*n])(b(n-1, m+1))+m*b(n-1, m))
end:
a:= n-> b(n-1, 0)[2]:
seq(a(n), n=2..24); # Alois P. Heinz, Dec 15 2023
MATHEMATICA
a[n_] := (n+1) BellB[n] - BellB[n+1];
Table[a[n], {n, 2, 24}] (* Jean-François Alcover, Dec 01 2018 *)
PROG
(Python)
# First version, simple recursion
from sympy import bell
HOW_MANY = 30
print([(n + 1) * bell(n) - bell(n + 1) for n in range(HOW_MANY)])
(Python)
# Second version by Taylor expansion
from sympy import *
from sympy.abc import z
bell = exp( exp (z) - 1)
h = (z * exp (z) - exp (z) + 1) * bell
NUMBER_OF_COEFFS = 8
coeffs = Poly(series(h, n = NUMBER_OF_COEFFS)).coeffs()
coeffs.reverse()
# and remove first coefficient 1 that corresponds to O(n**k)
coeffs.pop(0)
print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))])
CROSSREFS
KEYWORD
nonn
AUTHOR
Sergey Kirgizov, Nov 26 2016
STATUS
approved