A332800 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.

1, 1, 2, 4, 9, 21, 44, 109, 241, 530, 1176, 3180, 6456, 14835, 34672, 81877, 179434, 479275, 977224, 2503363, 5339049, 11207391, 28379591, 82473713, 166689486, 370775384, 877910547, 2150475950, 4608590865, 12146671367, 24620749285, 64137229920, 143062854926

Offset: 0

Author

Seiichi Manyama, Feb 27 2020

Keywords

• nonn

Comments

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.

Examples

			b(n) = sigma(n) mod sigma(n+1).
In case of n = 3.
    |           | b(1),b(2)
----+-----------+----------
  1 | [1, 2, 3] | [1, 2] *
  2 | [1, 3, 2] | [1, 1]
  3 | [2, 1, 3] | [0, 1] *
  4 | [3, 1, 2] | [0, 1] *
In case of n = 4.
    |              | b(1),b(2),b(3)
----+--------------+---------------
  1 | [1, 2, 3, 4] | [1, 2, 3] *
  2 | [1, 3, 2, 4] | [1, 1, 2]
  3 | [1, 4, 3, 2] | [1, 1, 1]
  4 | [2, 1, 3, 4] | [0, 1, 3] *
  5 | [2, 1, 4, 3] | [0, 1, 1]
  6 | [3, 1, 2, 4] | [0, 1, 2] *
  7 | [4, 1, 2, 3] | [0, 1, 2] *
  8 | [4, 1, 3, 2] | [0, 1, 1]
  9 | [4, 2, 1, 3] | [0, 0, 1]
* (strongly increasing)
		

Crossrefs

Cf. A022825.

Extensions

a(17)-a(20) from Alois P. Heinz, Feb 27 2020

a(21)-a(22) from Giovanni Resta, Mar 03 2020

a(23)-a(31) from Bert Dobbelaere, Mar 12 2020

a(32) from Bert Dobbelaere, Mar 15 2020