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, 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

Author

Marcel K. Goh, Dec 23 2020

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• M. K. Goh and R. Y. Zhao, Arithmetic subsequences in a random ordering of an additive set, arXiv:2012.12339 [math.CO], 2020.

Crossrefs

Cf. A000010, A002618, A339941.

Formula

T(n,n) = n*A000010(n).