A243676 Number of hypoplactic classes of 4-parking functions of length n.

1, 1, 9, 113, 1649, 26225, 440985, 7711009, 138792929, 2554489505, 47854963881, 909495557393, 17492724268369, 339846019830673, 6659441891042105, 131467175048437569, 2612224160086781889, 52201209713045788737, 1048450942860766632777, 21153308764742204273329, 428520989167282737342513

Offset: 0

Author

N. J. A. Sloane, Jun 14 2014

Keywords

• nonn

Comments

This is almost certainly the sequence of small 5-Schroeder numbers as defined by Yang-Jiang (2021). It would be nice to have a proof. Then we could confirm Weiner's conjectured formulas, 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 indeed the small 5-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

Appears to equal A260332(n)/2 for n > 0. - N. J. A. Sloane, Mar 28 2021

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

Programs

Maple

a := proc(n) option remember; if n <= 1 then return 1 fi;
-(a(n-2)*(-5440*n^7 + 42080*n^6 - 131548*n^5 + 212750*n^4 - 189160*n^3 + 90725*n^2 - 21387*n + 1890)+ a(n-1)*(-118660*n^7 + 739880*n^6 - 1876702*n^5 + 2492120*n^4 - 1855960*n^3 + 768230*n^2 - 161913*n + 13230)) / (5440*n^7 - 25760*n^6 + 43468*n^5 - 29510*n^4 + 4750*n^3 + 1945*n^2 - 468*n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Apr 13 2024

Mathematica

a[n_] := Hypergeometric2F1[1 - n, -4 n, 2, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 13 2024 *)

Formula

a(n+1) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(4*i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - Michael D. Weiner, May 25 2017

a(n) = Sum_{i=1..n} binomial(4*n, i-1)*binomial(n, i)*2^(i-1)/n (conjectured). - Michael D. Weiner, Jul 24 2019 [This is correct for n>0 - Jun Yan, Apr 13 2024]

Let D(n) be the set of 4-Dyck paths with n up-steps of size 4, 4n 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). - Jun Yan, Apr 13 2024

a(n) = hypergeom([1 - n, -4*n], [2], 2). - Peter Luschny, Apr 13 2024

Extensions

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

More terms from Jun Yan, Apr 13 2024