A309807 -id:A309807

Search: a309807 -id:a309807

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A332955

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

See A309807.

LINKS

Table of n, a(n) for n=0..33.

PROG

(Ruby)

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)

CROSSREFS

Cf. A309807.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 04 2020

EXTENSIONS

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

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

STATUS

approved

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.

+10
3

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

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.

LINKS

Seiichi Manyama, Rows n = 0..18, flattened

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

EXAMPLE

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;

CROSSREFS

T(n,0) gives A309807.

Cf. A052955.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Mar 04 2020

STATUS

approved

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.

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,13

COMMENTS

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.

LINKS

Seiichi Manyama, Rows n = 1..18, flattened

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

EXAMPLE

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;

CROSSREFS

Row sums give A309807.

Cf. A332954.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Mar 14 2020

STATUS

approved

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