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A339941 Triangle read by rows: T(n,k) is the number of permutations of {1,...,n} whose longest embedded arithmetic progression has length k.

%I A339941 #14 Feb 01 2021 13:20:23
%S A339941 1,0,2,0,4,2,0,10,12,2,0,20,82,16,2,0,48,516,134,20,2,0,104,3232,1480,
%T A339941 198,24,2,0,282,21984,15702,2048,274,28,2,0,496,168368,162368,28048,
%U A339941 3204,362,32,2,0,1066,1306404,1902496,374194,39420,4720,462,36,2,0,2460,11064306,23226786,4929828,622140,64020,6644,574,40,2
%N A339941 Triangle read by rows: T(n,k) is the number of permutations of {1,...,n} whose longest embedded arithmetic progression has length k.
%C A339941 Asymptotics can be found in Goh and Zhao (2020). The column k=2 corresponds to the number of 3-free permutations of 1..n, for n>=2.
%H A339941 M. K. Goh and R. Y. Zhao, <a href="https://arxiv.org/abs/2012.12339">Arithmetic subsequences in a random ordering of an additive set</a>, arXiv:2012.12339 [math.CO], 2020.
%e A339941 Triangle T(n,k) begins:
%e A339941   n/k 1    2         3         4        5       6      7     8    9 10 11 12
%e A339941    1  1
%e A339941    2  0    2
%e A339941    3  0    4         2
%e A339941    4  0   10        12         2
%e A339941    5  0   20        82        16        2
%e A339941    6  0   48       516       134       20       2
%e A339941    7  0  104      3232      1480      198      24      2
%e A339941    8  0  282     21984     15702     2048     274     28     2
%e A339941    9  0  496    168368    162368    28048    3204    362    32    2
%e A339941   10  0 1066   1306404   1902496   374194   39420   4720   462   36   2
%e A339941   11  0 2460  11064306  23226786  4929828  622140  64020  6644  574  40  2
%e A339941   12  0 6128 101355594 298314654 68584052 9719492 913440 98472 9024 698 44 2
%Y A339941 Cf. A003407 (column k=2), A338993, A339942.
%K A339941 nonn,tabl
%O A339941 1,3
%A A339941 _Marcel K. Goh_, Dec 23 2020