A370832 Triangle read by rows: T(n,k) gives the number of parking functions of size n with k lucky cars. 0 <= k <= n.

1, 0, 1, 0, 1, 2, 0, 2, 8, 6, 0, 6, 37, 58, 24, 0, 24, 204, 504, 444, 120, 0, 120, 1318, 4553, 6388, 3708, 720, 0, 720, 9792, 44176, 87296, 81136, 33984, 5040, 0, 5040, 82332, 463860, 1203921, 1582236, 1064124, 341136, 40320, 0, 40320, 773280, 5270480, 17164320, 29724000, 28328480, 14602320, 3733920, 362880

Offset: 0

Author

Peter Kagey, Mar 02 2024

Keywords

• nonn
• tabl

Comments

A car is called "lucky" if it gets its preferred parking spot.

Closely related to A220884.

Examples

			Table begins:
n\k|  0     1     2      3       4       5       6      7     8
---+-------------------------------------------------------------
 0 |  1
 1 |  0     1
 2 |  0     1     2
 3 |  0     2     8      6
 4 |  0     6    37     58      24
 5 |  0    24   204    504     444     120
 6 |  0   120  1318   4553    6388    3708     720
 7 |  0   720  9792  44176   87296   81136   33984   5040
 8 |  0  5040 82332 463860 1203921 1582236 1064124 341136 40320
      ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, Probabilistic Parking Functions, arXiv:2211.00536 [math.CO], 2022.
• FindStat, St000135: The number of lucky cars of the parking function.

Crossrefs

Cf. A002538, A067948, A220884.

Row sums give A000272(n+1).

Cf. A000142 (main diagonal and column k=1 shifted).

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      expand(x*mul((n+1-k)+k*x, k=2..n)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 26 2024

Mathematica

row[n_] := (x (x - 1)^n Pochhammer[(n + x) / (x - 1), n]) / (n + x);
Table[CoefficientList[Series[row[n], {x, 0, n}], x], {n, 0, 8}] // Flatten
(* Peter Luschny, Jun 27 2024 *)

Formula

T(n, n) = n!.

T(n, 1) = (n-1)!.

Sum_{k=1..n} T(n, k) = (n+1)^(n-1).

T(n+1, n) = A002538(n).

G.f. for row n>0: x * Product_{j=2..n} (n + 1 + j*(x-1)).

T(n, k) = [x^k] (x*(x - 1)^n*Pochhammer((n + x) / (x - 1), n)) / (n + x). - Peter Luschny, Jun 27 2024

Extensions

Edited by Alois P. Heinz, Jun 26 2024