1,3

For n >= 4, this is the number of permutations that avoid substrings j(j+3), 1 <= j <= n-3.

For n>=4, the number of circular permutations (in cycle notation) on [n+1] that avoid substrings (j,j+4), 1<=j<=n-3. For example, for n=4, there are 18 circular permutations in S5 that avoid the substring {15}. Note that each of these circular permutations represent 5 permutations in one-line notation (see link 2017). - Enrique Navarrete, Feb 22 2017

Indranil Ghosh, Table of n, a(n) for n = 1..400

Enrique Navarrete, Generalized K-Shift Forbidden Substrings in Permutations, arXiv:1610.06217 [math.CO], 2016.

Enrique Navarrete, Forbidden Substrings in Circular K-Successions, arXiv:1702.02637 [math.CO], 2017.

For n>=4: a(n) = Sum_{j=0..n-3} (-1)^j*binomial(n-3,j)*(n-j)!.

a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Oct 28 2016

a(5) = 78 since there are 78 permutations in S5 that avoid the substrings {14,25}.

Table[Sum[(-1)^j*Binomial[n - 3, j] (n - j)!, {j, 0, n - 3}], {n, 23}] (* Michael De Vlieger, Oct 27 2016 *)

Flatten[{0, 0, Table[n!*Hypergeometric1F1[3-n, -n, -1], {n, 3, 20}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

(PARI) a(n) = sum(j=0, n-3, (-1)^j*binomial(n-3, j)*(n-j)!); \\ Michel Marcus, Oct 29 2016

Cf. A068106, A000261.

Sequence in context: A332939 A299412 A218080 * A188119 A239420 A219590

Adjacent sequences: A277606 A277607 A277608 * A277610 A277611 A277612

Enrique Navarrete, Oct 23 2016

approved