|
%I #44 Jun 21 2021 01:55:12
%S 1,1,1,2,3,7,17,44,118,338,1003,3039,9466,30009,96757,316429,1047683,
%T 3511473,11876457,40537388,139490014,483393651,1686007017,5917253784,
%U 20879801881,74038098051,263793988890,943928231920,3390975927021,12227214763162,44242758258306
%N The width of the lattice of Dyck paths of length 2n ordered by the relation that one Dyck path lies above another one.
%C Previous name was: The width of the lattice E_n defined in the paper by Boldi and Vigna, that is, the cardinality of a maximal antichain.
%C a(n) is the maximum entry in row n of the triangle T(n,k) defined in A138158, or equivalently, the maximum entry in row n of the triangle T(n,k) defined in A227543. All level sizes of the lattice are given by A138158 and A227543. - _Torsten Muetze_, Nov 28 2018
%D Winston, Kenneth J., and Daniel J. Kleitman. "On the asymptotic number of tournament score sequences." Journal of Combinatorial Theory, Series A 35.2 (1983): 208-230. See Table 1.
%H Torsten Muetze, <a href="/A274291/b274291.txt">Table of n, a(n) for n = 0..300</a>
%H Paolo Boldi and Sebastiano Vigna, <a href="https://doi.org/10.1007/s11083-016-9418-8">On the Lattice of Antichains of Finite Intervals</a>, Order (2016), 1-25.
%H Paolo Boldi, Sebastiano Vigna, <a href="https://arxiv.org/abs/1510.03675">On the lattice of antichains of finite intervals</a>, arXiv preprint arXiv:1510.03675 [math.CO], 2015-2016.
%e For n=4 there are 14 Dyck paths, and 1,3,3,3,2,1,1 of them have area 0,1,2,3,4,5,6, respectively, where the area is normalized to the range 0,...,n(n-1)/2. These Dyck paths are UDUDUDUD (area=0), UUDDUDUD, UDUUDDUD, UDUDUUDD (area=1), UUDUDDUD, UDUUDUDD, UUDDUUDD (area=2), UUUDDDUD, UUDUDUDD, UDUUUDDD (area=3), UUUDDUDD, UUDUUDDD (area=4), UUUDUDDD (area=5), UUUUDDDD (area=6). The maximum among the numbers 1,3,3,3,2,1,1 is 3, so a(4)=3.
%Y Cf. A138158, A227543.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Jun 17 2016
%E a(0)=1 inserted by _Sebastiano Vigna_, Dec 20 2017
%E New name and more terms from _Torsten Muetze_, Nov 28 2018
|
