A292523 Decimal encoding T(n,k) of the k-th non-averaging permutation of [n]; triangle T(n,k), n >= 0, k = 1..A003407(n), read by rows.

0, 1, 12, 21, 132, 213, 231, 312, 1324, 1342, 2143, 2413, 2431, 3124, 3142, 3412, 4213, 4231, 15324, 15342, 21453, 24153, 24315, 24351, 24513, 31254, 31524, 31542, 35124, 35142, 35412, 42153, 42315, 42351, 42513, 45213, 51324, 51342, 153264, 153426, 153462

Offset: 0

Author

Alois P. Heinz, Dec 08 2017

Keywords

• nonn
• tabf
• base

Comments

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

The encoding of the empty permutation () is 0. For positive n each element in the permutation is encoded using 1+floor(log_10(n)) = A055642(n) digits with leading 0's if necessary. Then all elements are concatenated.

All terms are in increasing order.

Examples

			Triangle T(n,k) begins:
:            0;
:            1;
:           12, 21;
:          132, 213, 231, 312;
:         1324, 1342, 2143, 2413,  2431,    3124,  3142, 3412, 4213, 4231;
:        15324, 15342, 21453, 24153,    ...,    42513, 45213, 51324, 51342;
:       153264, 153426, 153462, 153624, ..., 624153, 624315, 624351, 624513;
:      1532764, 1537264, 1537426,       ...,       7351462, 7351624, 7356124;
:     15327648, 15327684, 15372648,     ...,     84627351, 84672315, 84672351;
:    195327648, 195327684, 195372648,   ...,   915738462, 915783426, 915783462;
:   1090503020710060408,                ...,               10020608090401050703;
:  109050302110710060408,               ...,              1103070910010502060804;
: 10905031107021006041208,              ...,             120408100206110307090105;
		

Links

• Alois P. Heinz, Rows n = 0..14, flattened
• Eric Weisstein's World of Mathematics, Nonaveraging Sequence
• Wikipedia, Arithmetic progression
• Index entries related to non-averaging sequences

Crossrefs

Cf. A003407, A030299, A055642, A088370.

Programs

Maple

T:= proc(n) option remember; local b, l, c; b, l, c:=
      proc(s, p) local ok, i, j, k;
        if nops(s) = 0 then l:= [l[], parse(p)]
      else for j in s do ok, i, k:= true, j-1, j+1;
             while ok and i>0 and k<=n do ok, i, k:=
               not i in s xor k in s, i-1, k+1 od;
             `if`(ok, b(s minus {j}, cat(p, 0$(c-length(j)), j)), 0)
           od
        fi
      end, [], length(n); b({$1..n}, "0"): sort(l)[]
    end:
seq(T(n), n=0..6);