A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.

1, 2, 1, 8, 5, 3, 50, 34, 25, 16, 432, 307, 243, 189, 125, 4802, 3506, 2881, 2401, 1921, 1296, 65536, 48729, 40953, 35328, 30208, 24583, 16807, 1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144, 20000000, 15217031, 13119879, 11708091, 10546875, 9453125, 8291909, 6880121, 4782969

Offset: 1

Author

Rui Duarte, Jan 22 2018

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
        1;
        2,      1;
        8,      5,      3;
       50,     34,     25,     16;
      432,    307,    243,    189,    125;
     4802,   3506,   2881,   2401,   1921,   1296;
    65536,  48729,  40953,  35328,  30208,  24583,  16807;
  1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144;
  ...
		

Links

• Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 6.
• D. Foata and J. Riordan, Mappings of acyclic and parking functions, J. Aeq. Math., 10 (1974) 10-22.

Crossrefs

Cf. A000272, A298593, A298594, A298597.

Programs

Mathematica

Table[Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

Formula

T(n,k) = Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).

T(n,k) = A298593(n,k)/n.

T(n,k) = Sum_{j=k..n} A298594(n,j).

T(n,k) = (Sum_{j=k..n} A298597(n,j))/n.

Sum_{k=1..n} T(n,k) = A000272(n+1).