login
A374756
Triangle read by rows: T(n,k) is the number of parking functions of order n where the k-th car is lucky.
2
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).
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
KEYWORD
nonn,tabl
AUTHOR
Kimberly P. Hadaway, Jul 18 2024
STATUS
approved