A298594 Triangle read by rows: T(n,k) = number of parking functions a of length n such that a(1) = k and if we replace a(1) = k with k+1 we don't get a parking function.

1, 1, 1, 3, 2, 3, 16, 9, 9, 16, 125, 64, 54, 64, 125, 1296, 625, 480, 480, 625, 1296, 16807, 7776, 5625, 5120, 5625, 7776, 16807, 262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144, 4782969, 2097152, 1411788, 1161216, 1093750, 1161216, 1411788, 2097152, 4782969

Offset: 1

Author

Rui Duarte, Jan 22 2018

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
       1;
       1,      1;
       3,      2,     3;
      16,      9,     9,    16;
     125,     64,    54,    64,   125;
    1296,    625,   480,   480,   625,  1296;
   16807,   7776,  5625,  5120,  5625,  7776,  16807;
  262144, 117649, 81648, 70000, 70000, 81648, 117649, 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.

Crossrefs

Cf. A000272, A298592, A298593, A298597.

Programs

Mathematica

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

Formula

T(n,k) = binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).

T(n,k) = A298592(n,k) - A298592(n,k+1).

T(n,k) = (A298593(n,k) - A298593(n,k+1))/n.

T(n,k) = A298597(n,k)/n.

T(n,1) = A000272(n+2).

T(n,n) = A000272(n+2).

T(n,k) = T(n,n-k).