A298592
Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.
Original entry on oeis.org
1, 2, 1, 8, 5, 3, 50, 34, 25, 16, 432, 307, 243, 189, 125, 4802, 3506, 2881, 2401, 1921, 1296, 65536, 48729, 40953, 35328, 30208, 24583, 16807, 1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144, 20000000, 15217031, 13119879, 11708091, 10546875, 9453125, 8291909, 6880121, 4782969
Offset: 1
Triangle begins:
1;
2, 1;
8, 5, 3;
50, 34, 25, 16;
432, 307, 243, 189, 125;
4802, 3506, 2881, 2401, 1921, 1296;
65536, 48729, 40953, 35328, 30208, 24583, 16807;
1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144;
...
- 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. 6.
- D. Foata and J. Riordan, Mappings of acyclic and parking functions, J. Aeq. Math., 10 (1974) 10-22.
-
Table[Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)
A298593
Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.
Original entry on oeis.org
1, 4, 2, 24, 15, 9, 200, 136, 100, 64, 2160, 1535, 1215, 945, 625, 28812, 21036, 17286, 14406, 11526, 7776, 458752, 341103, 286671, 247296, 211456, 172081, 117649, 8503056, 6405904, 5464712, 4811528, 4251528, 3691528, 3038344, 2097152, 180000000, 136953279, 118078911, 105372819, 94921875, 85078125, 74627181, 61921089, 43046721
Offset: 1
Triangle begins:
====================================================================
n\k| 1 2 3 4 5 6 7 8
---|----------------------------------------------------------------
1 | 1
2 | 4 2
3 | 24 15 9
4 | 200 136 100 64
5 | 2160 1535 1215 945 625
6 | 28812 21036 17286 14406 11526 7776
7 | 458752 341103 286671 247296 211456 172081 117649
8 | 8503056 6405904 5464712 4811528 4251528 3691528 3038344 2097152
...
-
Table[n Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)
A298597
Number T(n,k) of times the value k appears on the parking functions of length n and such that if we replace that value k with k+1 we don't get a parking function.
Original entry on oeis.org
1, 2, 2, 9, 6, 9, 64, 36, 36, 64, 625, 320, 270, 320, 625, 7776, 3750, 2880, 2880, 3750, 7776, 117649, 54432, 39375, 35840, 39375, 54432, 117649, 2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152, 43046721, 18874368, 12706092, 10450944, 9843750, 10450944, 12706092, 18874368, 43046721
Offset: 1
Triangle begins:
1;
2, 2;
9, 6, 9;
64, 36, 36, 64;
625, 320, 270, 320, 625;
7776, 3750, 2880, 2880, 3750, 7776;
117649, 54432, 39375, 35840, 39375, 54432, 117649;
2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152;
...
-
Table[n Binomial[n - 1, k - 1] k^(k - 2)*(n + 1 - k)^(n - 1 - k), {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)
Showing 1-3 of 3 results.
Comments