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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.

%I A374756 #42 Dec 27 2024 08:55:22
%S A374756 1,3,2,16,11,9,125,87,74,64,1296,908,783,708,625,16807,11824,10266,
%T A374756 9421,8733,7776,262144,184944,161221,148992,140298,131632,4782969,
%U A374756 3381341,2955366,2742090,2600879,2480787,100000000,70805696,61999923,57671104,54921875,52779840,2357947691,1671605646,1465709426,1365730231,1303885965,1258181726
%N A374756 Triangle read by rows: T(n,k) is the number of parking functions of order n where the k-th car is lucky.
%C A374756 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).
%H A374756 Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See pp. 7, 10.
%e A374756 Triangle begins:
%e A374756       1;
%e A374756       3,     2;
%e A374756      16,    11,     9;
%e A374756     125,    87,    74,   64;
%e A374756    1296,   908,   783,  708,  625;
%e A374756   16807, 11824, 10266, 9421, 8733, 7776;
%e A374756   ...
%e A374756 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.
%Y A374756 Cf. A000169 (leading diagonal), A374533 (second diagonal).
%Y A374756 Columns k = 1..5: A000272, A372842, A372843, A372844, A372845.
%Y A374756 Cf. A370832.
%K A374756 nonn,tabl
%O A374756 1,2
%A A374756 _Kimberly P. Hadaway_, Jul 18 2024