A370832 -id:A370832 - cp's OEIS frontend

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

Kimberly P. Hadaway, Jul 18 2024

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).

			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.

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.

Cf. A000169 (leading diagonal), A374533 (second diagonal).
Columns k = 1..5: A000272, A372842, A372843, A372844, A372845.
Cf. A370832.

A375616 a(n) is the number of lucky cars in all parking functions of order n.

Original entry on oeis.org

0, 1, 5, 36, 350, 4320, 64827, 1146880, 23383404, 540000000, 13933327265, 397303087104, 12407264266410, 421154777645056, 15439814208984375, 607985949695016960, 25593429637028941208, 1146928904801167933440, 54515427164280400691709, 2739404800000000000000000

Offset: 0

Kimberly P. Hadaway, Aug 21 2024, suggested by Andrew Howroyd

nonn

This sequence enumerates lucky cars in parking functions of order n (where a lucky spot is one which is parked in by a car which prefers that spot).

Alois P. Heinz, Table of n, a(n) for n = 0..386
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 p. 10.

Row sums of A374756.

Cf. A370832.

b:= proc(n) option remember; `if`(n=0, 1,
      expand(x*mul((n+1-k)+k*x, k=2..n)))
    end:
a:= n-> add(k*coeff(b(n), x, k), k=1..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 21 2024

b[n_] := b[n] = If[n == 0, 1, Expand[x*Product[(n+1-k) + k*x, {k, 2, n}]]];
a[n_] := Sum[k*Coefficient[b[n], x, k], {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 31 2024, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k*A370832(n,k) = Sum_{k=1..n} A374756(n,k).

a(6)-a(19) from Alois P. Heinz, Aug 21 2024

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

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A375616 a(n) is the number of lucky cars in all parking functions of order n.

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