login
Number of hypoplactic classes of 5-parking functions of length n.
0

%I #25 Apr 13 2024 16:28:56

%S 1,1,11,171,3101,61381,1285663,28015735,628599577,14424917769,

%T 336976627571,7986962580515,191593388321973,4642728729231885,

%U 113479537427180871,2794487353521152111,69264992525510000817,1726686697658673068305,43263333673014161542363,1088922278007765403976219,27519721658072318988515021

%N Number of hypoplactic classes of 5-parking functions of length n.

%C See Novelli-Thibon (2014) for precise definition.

%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-202. See Fig. 23.

%H Jun Yan, <a href="http://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.

%F 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

%F From _Jun Yan_, Apr 13 2024: (Start)

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

%F 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)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 14 2014

%E More terms from _Jun Yan_, Apr 13 2024