- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000374: Permutations ⟶ ℤ

Values

0 => [2] => [1,1,0,0] => [1,2] => 0

1 => [1,1] => [1,0,1,0] => [2,1] => 1

00 => [3] => [1,1,1,0,0,0] => [1,2,3] => 0

01 => [2,1] => [1,1,0,0,1,0] => [1,3,2] => 1

10 => [1,2] => [1,0,1,1,0,0] => [2,1,3] => 1

11 => [1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 1

000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0

001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 1

010 => [2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 1

011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 1

100 => [1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 1

101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 2

110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 1

111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 1

0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0

0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 1

0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 1

0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 1

0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 1

0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 2

0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 1

0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 1

1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 1

1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 2

1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 2

1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 2

1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 1

1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 2

1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 1

1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1

00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0

00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 1

00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 1

00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 1

00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 1

00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 2

00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => 1

00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => 1

01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 1

01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 2

01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 2

01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 2

01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => 1

01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 2

01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => 1

01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => 1

10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 1

10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 2

10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 2

10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 2

10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 2

10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 3

10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 2

10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 2

11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 1

11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 2

11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 2

11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 2

11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 1

11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 2

11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 1

11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 1

000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0

000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => 1

000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,2,3,4,6,5,7] => 1

000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,2,3,4,6,7,5] => 1

000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,2,3,5,4,6,7] => 1

000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,2,3,5,4,7,6] => 2

000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,2,3,5,6,4,7] => 1

000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,2,3,5,6,7,4] => 1

001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,2,4,3,5,6,7] => 1

001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,2,4,3,5,7,6] => 2

001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5,7] => 2

001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,2,4,3,6,7,5] => 2

001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,2,4,5,3,6,7] => 1

001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,2,4,5,3,7,6] => 2

001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,2,4,5,6,3,7] => 1

001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,2,4,5,6,7,3] => 1

010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,3,2,4,5,6,7] => 1

010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,3,2,4,5,7,6] => 2

010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2,4,6,5,7] => 2

010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,3,2,4,6,7,5] => 2

010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,5,4,6,7] => 2

010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,7,6] => 3

010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,5,6,4,7] => 2

010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,5,6,7,4] => 2

011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,3,4,2,5,6,7] => 1

011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,3,4,2,5,7,6] => 2

011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,3,4,2,6,5,7] => 2

011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,3,4,2,6,7,5] => 2

011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,3,4,5,2,6,7] => 1

011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,3,4,5,2,7,6] => 2

011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,3,4,5,6,2,7] => 1

011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,7,2] => 1

100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => 1

110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,1,4,5,6,7] => 1

111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,4,1,5,6,7] => 1

111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,4,1,5,7,6] => 2

111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,1,6,7] => 1

111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,5,1,7,6] => 2

111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,1,7] => 1

>>> Load all 140 entries. <<<

111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1] => 1

0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => 0

0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [1,2,3,4,5,6,8,7] => 1

0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [1,2,3,4,5,7,6,8] => 1

0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [1,2,3,5,4,6,7,8] => 1

0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6,8] => 2

0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0] => [1,2,3,5,6,7,8,4] => 1

0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,6,5,7,8] => 2

0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,3,2,4,5,6,7,8] => 1

0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,3,2,4,5,7,6,8] => 2

0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,5,4,6,7,8] => 2

0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6,8] => 3

0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,6,7,5,8] => 2

1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,1,3,4,5,6,7,8] => 1

1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,1,3,4,5,6,8,7] => 2

1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5,8,7] => 4

1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,1,4,5,7,8,6] => 2

1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [2,3,4,1,6,7,8,5] => 2

1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1,6,7,8] => 1

1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,6,1,7,8] => 1

1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,5,6,1,8,7] => 2

1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,1,8] => 1

1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,1] => 1

00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9] => 0

00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [1,2,3,4,5,6,7,9,8] => 1

10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,1,3,4,5,6,7,8,9] => 1

11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,5,6,7,1,8,9] => 1

11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,8,1,9] => 1

11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,9,1] => 1

000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9,10] => 0

000000001 => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [1,2,3,4,5,6,7,8,10,9] => 1

100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,1,3,4,5,6,7,8,9,10] => 1

101010101 => [1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5,8,7,10,9] => 5

111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,6,7,8,9,1,10] => 1

111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,8,9,10,1] => 1

1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,1,3,4,5,6,7,8,9,10,11] => 1

=> [1] => [1,0] => [1] => 0

0000000001 => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [1,2,3,4,5,6,7,8,9,11,10] => 1

10101010101 => [1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6

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$F_{1} = 1 + q$

$F_{2} = 1 + 3\ q$

$F_{3} = 1 + 6\ q + q^{2}$

$F_{4} = 1 + 10\ q + 5\ q^{2}$

$F_{5} = 1 + 15\ q + 15\ q^{2} + q^{3}$

Description

The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ .
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.

Map

bounce path

Description

The bounce path determined by an integer composition.