A274291 The width of the lattice of Dyck paths of length 2n ordered by the relation that one Dyck path lies above another one.

1, 1, 1, 2, 3, 7, 17, 44, 118, 338, 1003, 3039, 9466, 30009, 96757, 316429, 1047683, 3511473, 11876457, 40537388, 139490014, 483393651, 1686007017, 5917253784, 20879801881, 74038098051, 263793988890, 943928231920, 3390975927021, 12227214763162, 44242758258306

Offset: 0

Author

N. J. A. Sloane, Jun 17 2016

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

References

• 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.

Links

• Torsten Muetze, Table of n, a(n) for n = 0..300
• Paolo Boldi and Sebastiano Vigna, On the Lattice of Antichains of Finite Intervals, Order (2016), 1-25.
• Paolo Boldi, Sebastiano Vigna, On the lattice of antichains of finite intervals, arXiv preprint arXiv:1510.03675 [math.CO], 2015-2016.

Crossrefs

Cf. A138158, A227543.

Extensions

a(0)=1 inserted by Sebastiano Vigna, Dec 20 2017

New name and more terms from Torsten Muetze, Nov 28 2018