A394237 Number of circular parking functions of length n avoiding the pattern 122.

1, 1, 2, 5, 19, 101, 676, 5377, 49295, 510729, 5894110, 74915741, 1039180187, 15613569325, 252501251744, 4371586879129, 80652138666871, 1579212732426257, 32701859350855770, 713914404925713589, 16384896394304282723, 394340620941231415541, 9929838681717090607612

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Lara Pudwell, Pattern avoidance in cyclic parking functions, 2025.

Lara Pudwell, Pattern Avoidance in Circular Parking Functions, Valparaiso Univ. (2026). See p. 6 (Table 2).

Formula

a(n) = 1 + Sum_{k=1..n-1} (binomial(n-1, k)^2*k!/(n - k)).

a(n) = A350267(n-1) + 1 for n>0. - Hugo Pfoertner, Mar 14 2026

Example

For n=4, the a(4)=19 parking functions are 1111, 1112, 1113, 1114, 1123, 1124, 1132, 1134, 1142, 1143, 1213, 1214, 1234, 1243, 1314, 1324, 1342, 1423, 1432.

Maple

a := n -> local k; 1 + add(binomial(n - 1, k)^2*k!/(n - k), k = 1 .. n - 1):
seq(a(i), i = 0 .. 22);

Mathematica

Table[1 + Sum[(Binomial[n - 1, k]^2*k!/(n - k)), {k, n - 1}], {n, 0, 22}] (* Michael De Vlieger, Mar 13 2026 *)

Crossrefs

Cf. A019536, A121774, A171005, A213937, A350267, A393352, A394236, A394238.

Sequence in context: A020117 A054687 A076669 * A093502 A009311 A211207

Adjacent sequences: A394234 A394235 A394236 * A394238 A394239 A394240

Keyword

nonn

Author

Lara Pudwell, Mar 13 2026

Status

approved