- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001471: Dyck paths ⟶ ℤ (values match St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

0000 => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1

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

0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 2

0011 => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 2

0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => 2

0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 3

0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => 2

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

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

1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => 2

1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 3

1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => 2

1100 => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 2

1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 2

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

1111 => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1

00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1

00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 2

00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 2

00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 2

00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 2

00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 2

00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2

01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => 3

01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 2

01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 2

01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 2

01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 3

01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 2

01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2

10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2

10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0] => 2

10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 3

10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 2

10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 2

10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 2

10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => 3

10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => 2

11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 2

11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 2

11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 2

11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 2

11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 2

11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1

000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1

000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2

000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 2

000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2

000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3

000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2

000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2

001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3

001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 4

001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4

001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2

001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2

001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 2

001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2

010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 2

010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3

010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2

010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 4

010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 4

010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3

010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3

011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2

011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3

011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 3

011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3

011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2

011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3

011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2

011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2

100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2

100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3

100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2

100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3

100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 3

100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3

>>> Load all 254 entries. <<<

100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2

101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3

101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3

101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 4

101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 4

101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2

101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3

101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 2

110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2

110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 2

110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2

110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2

110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4

110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 4

110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3

110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2

111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2

111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3

111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2

111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 2

111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2

111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1

0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1

0000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2

0000010 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0] => 2

0000011 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 2

0000100 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => 3

0000101 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,1,0,1,0,0] => 3

0000110 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => 2

0000111 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 2

0001000 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 2

0001001 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => 2

0001010 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 2

0001011 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => 2

0001100 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0] => 3

0001101 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0] => 3

0001110 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 2

0001111 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 2

0010000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => 3

0010001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0,1,1,0,0] => 3

0010010 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0,1,0] => 3

0010011 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0] => 3

0010100 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => 3

0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => 3

0010110 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => 3

0010111 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => 2

0011000 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => 3

0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0] => 3

0011010 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3

0011011 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => 3

0011100 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 2

0011101 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 2

0011110 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => 2

0011111 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2

0100000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,1,0,0] => 2

0100001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => 3

0100010 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 2

0100011 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0] => 2

0100100 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => 3

0100101 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 3

0100110 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0] => 3

0100111 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0] => 3

0101000 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => 2

0101001 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => 3

0101010 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 3

0101011 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => 3

0101100 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0] => 3

0101101 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => 3

0101110 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0] => 3

0101111 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0] => 3

0110000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,0,1,1,1,0,0,0] => 2

0110001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0,1,1,0,0] => 3

0110010 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0] => 3

0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0,1,0] => 3

0110100 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0] => 3

0110101 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 3

0110110 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0] => 2

0110111 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0] => 2

0111000 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0] => 2

0111001 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,1,0,0,0] => 3

0111010 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => 3

0111011 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0] => 3

0111100 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => 2

0111101 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 3

0111110 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 2

0111111 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

1000000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2

1000001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 2

1000010 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 3

1000011 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => 2

1000100 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0] => 3

1000101 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => 3

1000110 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,1,0,0,0] => 3

1000111 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0] => 2

1001000 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,1,1,1,0,0,0,0] => 2

1001001 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0] => 2

1001010 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 3

1001011 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0] => 3

1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0,1,0] => 3

1001101 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0] => 3

1001110 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0,1,1,0,0] => 3

1001111 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,0,1,1,1,0,0,0] => 2

1010000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0] => 3

1010001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0] => 3

1010010 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => 3

1010011 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0] => 3

1010100 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => 3

1010101 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 3

1010110 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => 3

1010111 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => 2

1011000 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0] => 3

1011001 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0] => 3

1011010 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 3

1011011 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => 3

1011100 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0] => 2

1011101 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 2

1011110 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => 3

1011111 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,1,0,0] => 2

1100000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2

1100001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => 2

1100010 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 2

1100011 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 2

1100100 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => 3

1100101 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3

1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0] => 3

1100111 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => 3

1101000 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0] => 2

1101001 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0] => 3

1101010 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => 3

1101011 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => 3

1101100 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0] => 3

1101101 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0,1,0] => 3

1101110 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0,1,1,0,0] => 3

1101111 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => 3

1110000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 2

1110001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0,1,1,0,0] => 2

1110010 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0] => 3

1110011 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0] => 3

1110100 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => 2

1110101 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 2

1110110 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => 2

1110111 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 2

1111000 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 2

1111001 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => 2

1111010 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,1,0,1,0,0] => 3

1111011 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => 3

1111100 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 2

1111101 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0] => 2

1111110 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2

1111111 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1

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$F_{1} = 2\ q$

$F_{2} = 2\ q + 2\ q^{2}$

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

$F_{4} = 2\ q + 12\ q^{2} + 2\ q^{3}$

$F_{5} = 2\ q + 26\ q^{2} + 4\ q^{3}$

$F_{6} = 2\ q + 32\ q^{2} + 22\ q^{3} + 8\ q^{4}$

$F_{7} = 2\ q + 54\ q^{2} + 72\ q^{3}$

Description

The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word

$w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in

$w$ .

Map

Barnabei-Castronuovo involution

Description

The Barnabei-Castronuovo Schützenberger involution on Dyck paths.
The image of a Dyck path is obtained by reversing the canonical decompositions of the two halves of the Dyck path. More precisely, let

$D_1, 1, D_2, 1, \dots$ be the canonical decomposition of the first half, then the canonical decomposition of the first half of the image is

$\dots, 1, D_2, 1, D_1$ .