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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.
1
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, 210, 42, 0, 128, 4192, 27872, 6592, 1312, 192, 32, 0, 0, 57402, 182790, 99630, 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
OFFSET
1,3
COMMENTS
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.
LINKS
M. K. Goh and R. Y. Zhao, Arithmetic subsequences in a random ordering of an additive set, arXiv:2012.12339 [math.CO], 2020.
FORMULA
T(n,n) = n*A000010(n).
EXAMPLE
Triangle T(n,k) begins:
n/k 1 2 3 4 5 6 7 8 9 10 11
1 1
2 0 2
3 0 0 6
4 0 8 8 8
5 0 0 40 60 20
6 0 0 468 192 48 12
7 0 0 462 3150 1176 210 42
8 0 128 4192 27872 6592 1312 192 32
9 0 0 57402 182790 99630 19656 2970 378 54
10 0 0 67440 1795320 1594640 146200 22000 2840 320 40
11 0 0 61050 17433130 17373620 4289340 662860 85910 9790 990 110
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Marcel K. Goh, Dec 23 2020
STATUS
approved