login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296529 Number T(n,k) of non-averaging permutations of [n] with first element k; triangle T(n,k), n >= 0, k = 0..n, read by rows. 4
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 3, 3, 2, 0, 2, 5, 6, 5, 2, 0, 5, 6, 13, 13, 6, 5, 0, 10, 10, 16, 32, 16, 10, 10, 0, 28, 26, 36, 51, 51, 36, 26, 28, 0, 24, 50, 62, 74, 76, 74, 62, 50, 24, 0, 50, 50, 134, 138, 161, 161, 138, 134, 50, 50, 0, 124, 120, 146, 302, 345, 386, 345, 302, 146, 120, 124 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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

T(0,0) = 1 by convention.

LINKS

Alois P. Heinz, Rows n = 0..99, flattened

Eric Weisstein's World of Mathematics, Nonaveraging Sequence

Wikipedia, Arithmetic progression

Index entries related to non-averaging sequences

FORMULA

T(n,k) = T(n,n+1-k) > 0 for k=1..n.

EXAMPLE

T(5,1) = 2: 15324, 15342.

T(5,2) = 5: 21453, 24153, 24315, 24351, 24513.

T(5,3) = 6: 31254, 31524, 31542, 35124, 35142, 35412.

T(5,4) = 5: 42153, 42315, 42351, 42513, 45213.

T(5,5) = 2: 51324, 51342.

Triangle T(n,k) begins:

1;

0, 1;

0, 1, 1;

0, 1, 2, 1;

0, 2, 3, 3, 2;

0, 2, 5, 6, 5, 2;

0, 5, 6, 13, 13, 6, 5;

0, 10, 10, 16, 32, 16, 10, 10;

0, 28, 26, 36, 51, 51, 36, 26, 28;

0, 24, 50, 62, 74, 76, 74, 62, 50, 24;

0, 50, 50, 134, 138, 161, 161, 138, 134, 50, 50;

...

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<n do ok, i, k:=

not i in s xor k in s, i-1, k+1 od;

r:= r+ `if`(ok, b(s minus {j}), 0)

od; r

fi

end:

T:= (n, k)-> `if`(k=0, 0^n, b({$0..n} minus {k-1})):

seq(seq(T(n, k), k=0..n), n=0..14);

MATHEMATICA

b[s_List] := 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]];

T[0, 0]=1; T[n_, k_] := If[k==0, 0^n, b[Range[0, n] ~Complement~ {k-1}]];

Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2017, after Alois P. Heinz *)

CROSSREFS

Columns k=0-1 give: A000007, A296530 (for n>0).

Row sums give A003407.

T(n,n) gives A296530.

T(n,ceiling(n/2)) gives A296531.

Cf. A292523.

Sequence in context: A127597 A167749 A104770 * A110280 A061009 A144106

Adjacent sequences: A296526 A296527 A296528 * A296530 A296531 A296532

KEYWORD

nonn,tabl,look

AUTHOR

Alois P. Heinz, Dec 14 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 09:46 EST 2022. Contains 358700 sequences. (Running on oeis4.)