A332800 - OEIS

%I #52 Mar 15 2020 05:53:32

%S 1,1,2,4,9,21,44,109,241,530,1176,3180,6456,14835,34672,81877,179434,

%T 479275,977224,2503363,5339049,11207391,28379591,82473713,166689486,

%U 370775384,877910547,2150475950,4608590865,12146671367,24620749285,64137229920,143062854926

%N Number of permutations sigma of [n] such that (sigma(k) mod sigma(k+1)) <= (sigma(k+1) mod sigma(k+2)) for 1 <= k <= n - 2.

%C Conjecture: Number of permutations sigma such that (sigma(k) mod sigma(k+1)) < (sigma(k+1) mod sigma(k+2)) for 1 <= k <= n - 2 is equal to A022825(n). This is true for n <= 19.

%e b(n) = sigma(n) mod sigma(n+1).

%e In case of n = 3.

%e     |           | b(1),b(2)

%e ----+-----------+----------

%e   1 | [1, 2, 3] | [1, 2] *

%e   2 | [1, 3, 2] | [1, 1]

%e   3 | [2, 1, 3] | [0, 1] *

%e   4 | [3, 1, 2] | [0, 1] *

%e In case of n = 4.

%e     |              | b(1),b(2),b(3)

%e ----+--------------+---------------

%e   1 | [1, 2, 3, 4] | [1, 2, 3] *

%e   2 | [1, 3, 2, 4] | [1, 1, 2]

%e   3 | [1, 4, 3, 2] | [1, 1, 1]

%e   4 | [2, 1, 3, 4] | [0, 1, 3] *

%e   5 | [2, 1, 4, 3] | [0, 1, 1]

%e   6 | [3, 1, 2, 4] | [0, 1, 2] *

%e   7 | [4, 1, 2, 3] | [0, 1, 2] *

%e   8 | [4, 1, 3, 2] | [0, 1, 1]

%e   9 | [4, 2, 1, 3] | [0, 0, 1]

%e * (strongly increasing)

%Y Cf. A022825.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 27 2020

%E a(17)-a(20) from _Alois P. Heinz_, Feb 27 2020

%E a(21)-a(22) from _Giovanni Resta_, Mar 03 2020

%E a(23)-a(31) from _Bert Dobbelaere_, Mar 12 2020

%E a(32) from _Bert Dobbelaere_, Mar 15 2020