A243677 Number of hypoplactic classes of 5-parking functions of length n.

1, 1, 11, 171, 3101, 61381, 1285663, 28015735, 628599577, 14424917769, 336976627571, 7986962580515, 191593388321973, 4642728729231885, 113479537427180871, 2794487353521152111, 69264992525510000817, 1726686697658673068305, 43263333673014161542363, 1088922278007765403976219, 27519721658072318988515021

Offset

0,3

Comments

See Novelli-Thibon (2014) for precise definition.

Links

Table of n, a(n) for n=0..20.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-202. See Fig. 23.

Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.

Formula

a(n) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(5i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - Michael D. Weiner, May 25 2017

From Jun Yan, Apr 13 2024: (Start)

a(n) = (1/n) * Sum_{k=1..n} binomial(5*n, k - 1) * binomial(n, k)*2^(k - 1) for n > 0.

Let D(n) be the set of 5-Dyck paths that have n up-steps of size 5 and 5n down-steps of size 1 and never go below the x-axis. For every d in D(n), let peak(d) be the number of peaks in d. Then a(n) = Sum_{d in D(n)} 2^(peak(d) - 1). (End)

Crossrefs

Sequence in context: A064182 A139792 A025758 * A307168 A141955 A133243

Adjacent sequences: A243674 A243675 A243676 * A243678 A243679 A243680

Keyword

nonn,changed

Author

N. J. A. Sloane, Jun 14 2014

Extensions

More terms from Jun Yan, Apr 13 2024

Status

approved