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%I #38 Jul 25 2024 14:49:00
%S 1,3,2,16,11,9,125,87,74,64,1296,908,783,708,625,16807,11824,10266,
%T 9421,8733,7776,262144,184944,161221,148992,140298,131632,4782969,
%U 3381341,2955366,2742090,2600879,2480787,100000000,70805696,61999923,57671104,54921875,52779840,2357947691,1671605646,1465709426,1365730231,1303885965,1258181726
%N Triangle read by rows: T(n,k) is the number of parking functions of order n where the k-th car is lucky.
%C 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).
%e Triangle begins:
%e 1;
%e 3, 2;
%e 16, 11, 9;
%e 125, 87, 74, 64;
%e 1296, 908, 783, 708, 625;
%e 16807, 11824, 10266, 9421, 8733, 7776;
%e ...
%e 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 Cf. A000169 (leading diagonal), A374533 (second diagonal).
%Y Columns k = 1..5: A000272, A372842, A372843, A372844, A372845.
%Y Cf. A370832.
%K nonn,tabl
%O 1,2
%A _Kimberly P. Hadaway_, Jul 18 2024
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