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A339942 Triangle read by rows: T(n,k) is the number of permutations of the cyclic group Z/nZ whose longest embedded arithmetic progression has length k.

%I A339942 #18 Feb 01 2021 13:20:54
%S A339942 1,0,2,0,0,6,0,8,8,8,0,0,40,60,20,0,0,468,192,48,12,0,0,462,3150,1176,
%T A339942 210,42,0,128,4192,27872,6592,1312,192,32,0,0,57402,182790,99630,
%U A339942 19656,2970,378,54,0,0,67440,1795320,1594640,146200,22000,2840,320,40,0,0,61050,17433130,17373620,4289340,662860,85910,9790,990,110
%N A339942 Triangle read by rows: T(n,k) is the number of permutations of the cyclic group Z/nZ whose longest embedded arithmetic progression has length k.
%C A339942 For the case k=2, it can be proved that if n is a power of 2, then T(n,2)=2^{n-1}; otherwise T(n,2)=0 (Lemma 8 of Goh and Zhao (2020)). It can also be shown that T(n,n) = n*phi(n), where phi is the Euler totient function.
%H A339942 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.
%F A339942 T(n,n) = n*A000010(n).
%e A339942 Triangle T(n,k) begins:
%e A339942   n/k 1   2        3         4         5        6      7     8    9  10  11
%e A339942    1  1
%e A339942    2  0   2
%e A339942    3  0   0        6
%e A339942    4  0   8        8         8
%e A339942    5  0   0       40        60        20
%e A339942    6  0   0      468       192        48       12
%e A339942    7  0   0      462      3150      1176      210     42
%e A339942    8  0 128     4192     27872      6592     1312    192    32
%e A339942    9  0   0    57402    182790     99630    19656   2970   378   54
%e A339942   10  0   0    67440   1795320   1594640   146200  22000  2840  320  40
%e A339942   11  0   0    61050  17433130  17373620  4289340 662860 85910 9790 990 110
%Y A339942 Cf. A000010, A002618, A339941.
%K A339942 nonn,tabl
%O A339942 1,3
%A A339942 _Marcel K. Goh_, Dec 23 2020