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A370832 Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 1 <= k <= n. 0
1, 1, 2, 2, 8, 6, 6, 37, 58, 24, 24, 204, 504, 444, 120, 120, 1318, 4553, 6388, 3708, 720, 720, 9792, 44176, 87296, 81136, 33984, 5040, 5040, 82332, 463860, 1203921, 1582236, 1064124, 341136, 40320, 40320, 773280, 5270480, 17164320, 29724000, 28328480, 14602320, 3733920, 362880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A car is called "lucky" if it gets its preferred parking spot.
Closely related to A220884.
LINKS
Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, Probabilistic Parking Functions, arXiv:2211.00536 [math.CO], 2022.
FORMULA
T(n,n) = n!.
T(n,1) = (n-1)!.
Sum_{k=1..n} T(n,k) = (n+1)^(n-1).
Conjecture: T(n,n-1) = A002538(n-1).
Empirical g.f. for row n: Product_{j=2..n} x*(n + 1 + j*(x-1)).
EXAMPLE
Table begins:
n\k| 1 2 3 4 5 6 7 8
---+-----------------------------------------------------------
1 | 1
2 | 1 2
3 | 2 8 6
4 | 6 37 58 24
5 | 24 204 504 444 120
6 | 120 1318 4553 6388 3708 720
7 | 720 9792 44176 87296 81136 33984 5040
8 | 5040 82332 463860 1203921 1582236 1064124 341136 40320
CROSSREFS
Sequence in context: A371937 A119999 A061828 * A086535 A344601 A067436
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Mar 02 2024
STATUS
approved

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Last modified June 16 10:55 EDT 2024. Contains 373425 sequences. (Running on oeis4.)