A296530 Number of non-averaging permutations of [n] with first element n.

1, 1, 1, 1, 2, 2, 5, 10, 28, 24, 50, 124, 283, 528, 1266, 3715, 10702, 8740, 15414, 31988, 68465, 160964, 380124, 890738, 2230219, 3990852, 8354276, 20281732, 46056920, 131289988, 349369117, 1054037937, 3081527146, 2440225484, 4201202020, 7475926894, 13276918426

Offset: 0

Author

Alois P. Heinz, Dec 14 2017

Keywords

• nonn

Comments

A non-averaging permutation avoids any 3-term arithmetic progression.

a(0) = 1 by convention.

Examples

			a(4) = 2: 4213, 4231.
a(5) = 2: 51324, 51342.
a(6) = 5: 621453, 624153, 624315, 624351, 624513.
a(7) = 10: 7312564, 7315264, 7315426, 7315462, 7315624, 7351264, 7351426, 7351462, 7351624, 7356124.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..99
• Eric Weisstein's World of Mathematics, Nonaveraging Sequence
• Wikipedia, Arithmetic progression
• Index entries related to non-averaging sequences

Crossrefs

Main diagonal of A296529.

Cf. A003407, A292523.

Programs

Maple

b:= proc(s) option remember; local n, r, ok, i, j, k;
      if nops(s) = 1 then 1
    else n, r:= max(s), 0;
         for j in s minus {n} do ok, i, k:= true, j-1, j+1;
           while ok and i>=0 and k b({$0..n} minus {n-1}):
seq(a(n), n=0..30);

Mathematica

b[s_] := b[s] = Module[{n = Max[s], r = 0, ok, i, j, k}, If[Length[s] == 1, 1, Do[{ok, i, k} = {True, j - 1, j + 1}; While[ok && i >= 0 && k < n, {ok, i, k} = {FreeQ[s, i] ~Xor~ MemberQ[s, k], i - 1, k + 1}]; r = r + If[ok, b[s ~Complement~ {j}], 0], {j, s ~Complement~ {n}}]; r]];
a[n_] := b[Complement[Range[0, n], {n - 1}]]
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A296529(n,n).