A332954 -id:A332954 - cp's OEIS frontend

A309807 Number of permutations sigma of [n] such that sigma(k)/k > sigma(k+1)/(k+1) for 1 <= k <= n-1.

1, 1, 1, 2, 3, 6, 9, 19, 30, 60, 108, 222, 388, 874, 1601, 3244, 6437, 14056, 26545, 57326, 109333, 232751, 481137, 1002039, 1911740, 4261276, 8678424, 17734328, 36186279, 77402058, 154454851, 340848002, 691228119, 1460761640

Offset: 0

Seiichi Manyama, Mar 03 2020

a(n+1) is equal to the number of permutations sigma of [n] such that sigma(k)/k >= sigma(k+1)/(k+1) for 1 <= k <= n-1.

			In case of n = 3.
----+----------
  1 | [2, 3, 1]
  2 | [3, 2, 1]
In case of n = 4.
----+-------------
  1 | [2, 3, 4, 1]
  2 | [3, 4, 2, 1]
  3 | [4, 3, 2, 1]

Mathematics.StackExchange, Why are the numbers of two different permutations the same?, Mar 07 2020.

Row sums of A333310.

Cf. A330432, A332954.

def A(n)
  (1..n).to_a.permutation.select{|i| (1..n - 1).all?{|j| i[j - 1] * (j + 1) > i[j] * j}}.size
end
def A309807(n)
  (0..n).map{|i| A(i)}
end
p A309807(10)

a(19)-a(22) from Alois P. Heinz, Mar 03 2020
a(23)-a(25) from Giovanni Resta, Mar 04 2020
a(26)-a(33) from Bert Dobbelaere, Mar 15 2020

A333310 Triangle read by rows: T(n,k) is the number of permutations sigma of [n] such that sigma(1) = k and sigma(j)/j > sigma(j+1)/(j+1) for 1 <= j <= n-1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 2, 1, 3, 0, 1, 3, 5, 2, 3, 5, 0, 1, 3, 6, 5, 3, 4, 8, 0, 1, 4, 8, 12, 8, 5, 9, 13, 0, 1, 4, 12, 20, 18, 8, 11, 13, 21, 0, 1, 5, 18, 29, 42, 21, 22, 19, 27, 38, 0, 1, 5, 23, 44, 69, 48, 33, 30, 33, 38, 64

Offset: 1

Seiichi Manyama, Mar 14 2020

T(n+1,k+1) is equal to the number of permutations sigma of [n] such that sigma(1) = k and sigma(j)/j >= sigma(j+1)/(j+1) for 1 <= j <= n-1.

			Triangle begins:
n\k  | 1  2  3   4   5   6   7   8   9  10  11  12
-----+--------------------------------------------
   1 | 1;
   2 | 0, 1;
   3 | 0, 1, 1;
   4 | 0, 1, 1,  1;
   5 | 0, 1, 2,  1,  2;
   6 | 0, 1, 2,  2,  1,  3;
   7 | 0, 1, 3,  5,  2,  3,  5;
   8 | 0, 1, 3,  6,  5,  3,  4,  8;
   9 | 0, 1, 4,  8, 12,  8,  5,  9, 13;
  10 | 0, 1, 4, 12, 20, 18,  8, 11, 13, 21;
  11 | 0, 1, 5, 18, 29, 42, 21, 22, 19, 27, 38;
  12 | 0, 1, 5, 23, 44, 69, 48, 33, 30, 33, 38, 64;

Seiichi Manyama, Rows n = 1..18, flattened
Mathematics.StackExchange, Why are the numbers of two different permutations the same?, Mar 07 2020.

Row sums give A309807.

Cf. A332954.

cp's OEIS Frontend

A309807 Number of permutations sigma of [n] such that sigma(k)/k > sigma(k+1)/(k+1) for 1 <= k <= n-1.

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