Identifier
Values
0 => 1 => [1,1] => 2
1 => 0 => [2] => 1
00 => 01 => [2,1] => 2
01 => 10 => [1,2] => 2
10 => 11 => [1,1,1] => 3
11 => 00 => [3] => 1
000 => 001 => [3,1] => 2
001 => 010 => [2,2] => 2
010 => 101 => [1,2,1] => 2
011 => 100 => [1,3] => 2
100 => 011 => [2,1,1] => 3
101 => 110 => [1,1,2] => 3
110 => 111 => [1,1,1,1] => 4
111 => 000 => [4] => 1
0000 => 0001 => [4,1] => 2
0001 => 0010 => [3,2] => 2
0010 => 0101 => [2,2,1] => 2
0011 => 0100 => [2,3] => 2
0100 => 1001 => [1,3,1] => 2
0101 => 1010 => [1,2,2] => 2
0110 => 1011 => [1,2,1,1] => 3
0111 => 1000 => [1,4] => 2
1000 => 0011 => [3,1,1] => 3
1001 => 0110 => [2,1,2] => 3
1010 => 1101 => [1,1,2,1] => 3
1011 => 1100 => [1,1,3] => 3
1100 => 0111 => [2,1,1,1] => 4
1101 => 1110 => [1,1,1,2] => 4
1110 => 1111 => [1,1,1,1,1] => 5
1111 => 0000 => [5] => 1
00000 => 00001 => [5,1] => 2
00001 => 00010 => [4,2] => 2
00010 => 00101 => [3,2,1] => 2
00011 => 00100 => [3,3] => 2
00100 => 01001 => [2,3,1] => 2
00101 => 01010 => [2,2,2] => 2
00110 => 01011 => [2,2,1,1] => 3
00111 => 01000 => [2,4] => 2
01000 => 10001 => [1,4,1] => 2
01001 => 10010 => [1,3,2] => 2
01010 => 10101 => [1,2,2,1] => 2
01011 => 10100 => [1,2,3] => 2
01100 => 10011 => [1,3,1,1] => 3
01101 => 10110 => [1,2,1,2] => 3
01110 => 10111 => [1,2,1,1,1] => 4
01111 => 10000 => [1,5] => 2
10000 => 00011 => [4,1,1] => 3
10001 => 00110 => [3,1,2] => 3
10010 => 01101 => [2,1,2,1] => 3
10011 => 01100 => [2,1,3] => 3
10100 => 11001 => [1,1,3,1] => 3
10101 => 11010 => [1,1,2,2] => 3
10110 => 11011 => [1,1,2,1,1] => 3
10111 => 11000 => [1,1,4] => 3
11000 => 00111 => [3,1,1,1] => 4
11001 => 01110 => [2,1,1,2] => 4
11010 => 11101 => [1,1,1,2,1] => 4
11011 => 11100 => [1,1,1,3] => 4
11100 => 01111 => [2,1,1,1,1] => 5
11101 => 11110 => [1,1,1,1,2] => 5
11110 => 11111 => [1,1,1,1,1,1] => 6
11111 => 00000 => [6] => 1
=> => [1] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.
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.