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A027686 Number of ways to transform say (((((((ab)c)d)e)f)g)h) to (a(b(c(d(e(f(gh))))))) where there are n multiplications (hence n+1 variables) by repeatedly applying the one-way associative law ((xy)z) -> (x(yz)). 4

%I

%S 1,1,1,2,9,98,2981,340549,216569887,994441978397,36812710172987995,

%T 12001387004225881846755,37783429241635794906272195147,

%U 1255674108542254217846031366276646429,478743486470659944952229546087586449114251007,2262324605850021060149051111359520226936424091385392945

%N Number of ways to transform say (((((((ab)c)d)e)f)g)h) to (a(b(c(d(e(f(gh))))))) where there are n multiplications (hence n+1 variables) by repeatedly applying the one-way associative law ((xy)z) -> (x(yz)).

%C Number of maximal chains in the Tamari lattice T_n. For n=3 there are 2 maximal chains in the Tamari lattice T3, whose Hasse diagram is a pentagon. - _F. Chapoton_, Mar 15 2013

%D D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, see solution to Exercise 34.

%H Alois P. Heinz, <a href="/A027686/b027686.txt">Table of n, a(n) for n = 0..18</a>

%H Luke Nelson, <a href="https://doi.org/10.1016/j.disc.2016.11.030">A recursion on maximal chains in the Tamari lattices</a>, Discrete Mathematics 340.4 (2017): 661-677.

%H Luke Nelson, <a href="https://arxiv.org/abs/1709.02987">A recursion on maximal chains in the Tamari lattices</a>, arXiv:1709.02987 [math.CO], Sep 2017

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tamari_lattice">Tamari lattice</a>

%p s:= proc(n) s(n):=`if`(n=0, [], [s(n-1), []]) end:

%p f:= l-> l=[] or l[1]=[] and f(l[2]):

%p v:= proc(l) v(l):=`if`(f(l), [], [`if`(l[1]<>[],

%p [l[1][1], [l[1][2], l[2]]], [][]),

%p seq([w, l[2]], w=v(l[1])), seq([l[1], w], w=v(l[2]))])

%p end:

%p p:= proc(l) p(l):=`if`(f(l), 1, add(p(w), w=v(l))) end:

%p a:= n-> p(s(n)):

%p seq(a(n), n=0..10); # _Alois P. Heinz_, Mar 17 2013

%Y Row sums of A282698.

%Y Cf. A000108.

%K nonn

%O 0,4

%A _Don Knuth_

%E a(9)-a(14), a(15) from _Alois P. Heinz_, Mar 17 2013, Mar 27 2013

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Last modified August 2 02:53 EDT 2021. Contains 346409 sequences. (Running on oeis4.)