A374756 Triangle read by rows: T(n,k) is the number of parking functions of order n where the k-th car is lucky.

1, 3, 2, 16, 11, 9, 125, 87, 74, 64, 1296, 908, 783, 708, 625, 16807, 11824, 10266, 9421, 8733, 7776, 262144, 184944, 161221, 148992, 140298, 131632, 4782969, 3381341, 2955366, 2742090, 2600879, 2480787, 100000000, 70805696, 61999923, 57671104, 54921875, 52779840, 2357947691, 1671605646, 1465709426, 1365730231, 1303885965, 1258181726

Offset: 1

Author

Kimberly P. Hadaway, Jul 18 2024

Keywords

• nonn
• tabl

Comments

This sequence enumerates parking functions with n cars and n parking spots with lucky k-th spot (where a lucky spot is one which is parked in by a car which prefers that spot).

Examples

			Triangle begins:
      1;
      3,     2;
     16,    11,     9;
    125,    87,    74,   64;
   1296,   908,   783,  708,  625;
  16807, 11824, 10266, 9421, 8733, 7776;
  ...
For clarity, we write parentheses around parking functions. For n = 3 and k = n-1 = 2, the T(3,2) = 11 solutions are the parking functions of length 3 with a lucky second spot: (1,2,1),(1,2,2),(1,2,3),(1,3,2),(2,1,1),(2,1,2),(2,1,3),(2,2,1),(2,3,1),(3,1,2),(3,2,1). There are 5 parking functions of length 3 which do not have a lucky second spot: (1,1,1),(1,1,2),(1,1,3),(1,3,1),(3,1,1). For all of these, the car which parks in the second spot did not prefer the second spot; these parking functions do not contribute to our count.
		

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 pp. 7, 10.

Crossrefs

Cf. A000169 (leading diagonal), A374533 (second diagonal).

Columns k = 1..5: A000272, A372842, A372843, A372844, A372845.

Cf. A370832.