A243675 Number of hypoplactic classes of 3-parking functions of length n.

1, 1, 7, 67, 741, 8909, 113107, 1492103, 20251945, 280978681, 3967031839, 56811348235, 823250855181, 12049087175493, 177857857845675, 2644773866954255, 39581787842355409, 595745692419162737, 9011736489133233463, 136932249972928786387, 2089082351509217490613

Offset: 0

Author

N. J. A. Sloane, Jun 14 2014

Keywords

• nonn

Comments

See Novelli-Thibon (2014) for precise definition.

This is almost certainly the sequence of small 4-Schroeder numbers as defined by Yang-Jiang (2021). It would be nice to have a proof. Then we could confirm Weiner's conjectured formula, and extend the sequence. Yang & Jiang (2021) give an explicit formula for the small m-Schroeder numbers in Theorems 2.4 and 2.9. - N. J. A. Sloane, Mar 28 2021

This is also the small 4-Schroeder numbers defined by Yang and Jiang (2021) in Theorems 2.4 and 2.9. - Jun Yan, Apr 13 2024

References

• Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Links

• 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], 2014-2020. See Fig. 23.
• Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.

Crossrefs

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Apparently, a(n) = A144097/2, apart from the initial term. - N. J. A. Sloane, Mar 28 2021 [This is for n > 0 indeed the case. - Jun Yan, Apr 13 2024]

Programs

Maple

a := proc(n) option remember; if n <= 1 then return 1 fi;
-((945*n^5 - 5481*n^4 + 11685*n^3 - 11091*n^2 + 4470*n - 600)*a(n - 2) +
(-15610*n^5 + 67123*n^4 - 106824*n^3 + 77633*n^2 - 25514*n + 3000)*a(n - 1)) /
(945*n^5 - 2646*n^4 + 1731*n^3 + 294*n^2 - 204*n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Apr 13 2024

Mathematica

a[n_] := Hypergeometric2F1[1 - n, -3 n, 2, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 13 2024 *)

Formula

a(n) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(3i+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(3*n, k - 1) * binomial(n, k)*2^(k - 1) for n>0.

Let D(n) be the set of 3-Dyck paths with n up-steps of size 3, 3n 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)

a(n) = hypergeom([1 - n, -3*n], [2], 2). - Peter Luschny, Apr 13 2024

D-finite with recurrence -15*n*(3*n-1)*(3*n+1)*a(n) +(43*n^3+5403*n^2-8482*n+3228)*a(n-1) +6*(6039*n^3-33372*n^2+60401*n-35858) *a(n-2) +9*(-689*n^3+5938*n^2-17157*n+16616)*a(n-3) +27*(3*n-10)*(3*n-11)*(n-4)*a(n-4)=0. - R. J. Mathar, Apr 14 2024

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*(35*n^2-98*n+68)*a(n) +(-15610*n^5+67123*n^4-106824*n^3+77633*n^2-25514*n+3000)*a(n-1) +3*(n-2)*(3*n-4)*(3*n-5)*(35*n^2-28*n+5)*a(n-2)=0. - R. J. Mathar, Apr 14 2024

Extensions

Added a(0) = 1. - N. J. A. Sloane, Mar 28 2021

More terms from Jun Yan, Apr 13 2024