A309807 -id:A309807 - cp's OEIS frontend

A332955 a(0) = 1 and a(n) = A309807(n) - A309807(n-1) for n > 0.

1, 0, 0, 1, 1, 3, 3, 10, 11, 30, 48, 114, 166, 486, 727, 1643, 3193, 7619, 12489, 30781, 52007, 123418, 248386, 520902, 909701, 2349536, 4417148, 9055904, 18451951, 41215779, 77052793, 186393151, 350380117, 769533521

Offset: 0

Seiichi Manyama, Mar 04 2020

nonn

See A309807.

Cf. A309807.

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 A332955(n)
  a = (0..n).map{|i| A(i)}
  [1] + (1..n).map{|i| a[i] - a[i - 1]}
end
p A332955(10)

a(23)-a(25) from Giovanni Resta, Mar 04 2020

a(26)-a(33) from Bert Dobbelaere, Mar 15 2020

A332954 Triangle read by rows: T(n,k) is the number of permutations sigma of [n] such that sigma(j)/(j+k) > sigma(j+1)/(j+k+1) for 1 <= j <= n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 6, 3, 2, 1, 1, 1, 9, 5, 3, 2, 1, 1, 1, 19, 8, 5, 3, 2, 1, 1, 1, 30, 13, 7, 5, 3, 2, 1, 1, 1, 60, 21, 12, 7, 5, 3, 2, 1, 1, 1, 108, 38, 17, 11, 7, 5, 3, 2, 1, 1, 1

Offset: 0

Seiichi Manyama, Mar 04 2020

Conjecture: T(2*n+4,n) = A052955(n+2). This is true for n <= 10.

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

			Triangle begins:
n\k  |   0   1   2   3   4  5  6  7  8  9 10 11
-----+-----------------------------------------
   0 |   1;
   1 |   1,  1;
   2 |   1,  1,  1;
   3 |   2,  1,  1,  1;
   4 |   3,  2,  1,  1,  1;
   5 |   6,  3,  2,  1,  1, 1;
   6 |   9,  5,  3,  2,  1, 1, 1;
   7 |  19,  8,  5,  3,  2, 1, 1, 1;
   8 |  30, 13,  7,  5,  3, 2, 1, 1, 1;
   9 |  60, 21, 12,  7,  5, 3, 2, 1, 1, 1;
  10 | 108, 38, 17, 11,  7, 5, 3, 2, 1, 1, 1;
  11 | 222, 64, 31, 16, 11, 7, 5, 3, 2, 1, 1, 1;

Seiichi Manyama, Rows n = 0..18, flattened
Mathematics.StackExchange, Why are the numbers of two different permutations the same?, Mar 07 2020.

T(n,0) gives A309807.

Cf. A052955.

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

A332955 a(0) = 1 and a(n) = A309807(n) - A309807(n-1) for n > 0.

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A332954 Triangle read by rows: T(n,k) is the number of permutations sigma of [n] such that sigma(j)/(j+k) > sigma(j+1)/(j+k+1) for 1 <= j <= n-1.

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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.

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Author

Keywords

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Examples

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