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

1,2

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).

Links

Table of n, a(n) for n=1..45.

Example

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.

Crossrefs

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

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

Cf. A370832.

Sequence in context: A275463 A338280 A304989 * A055864 A209600 A072045

Adjacent sequences: A374752 A374753 A374755 * A374768 A374770 A374771

Keyword

nonn,tabl,new

Author

Kimberly P. Hadaway, Jul 18 2024

Status

approved