Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001658: Skew partitions ⟶ ℤ
Values
0 => [2] => [[2],[]] => 3
1 => [1,1] => [[1,1],[]] => 3
00 => [3] => [[3],[]] => 4
01 => [2,1] => [[2,2],[1]] => 5
10 => [1,2] => [[2,1],[]] => 5
11 => [1,1,1] => [[1,1,1],[]] => 4
000 => [4] => [[4],[]] => 5
001 => [3,1] => [[3,3],[2]] => 7
010 => [2,2] => [[3,2],[1]] => 8
011 => [2,1,1] => [[2,2,2],[1,1]] => 7
100 => [1,3] => [[3,1],[]] => 7
101 => [1,2,1] => [[2,2,1],[1]] => 8
110 => [1,1,2] => [[2,1,1],[]] => 7
111 => [1,1,1,1] => [[1,1,1,1],[]] => 5
0000 => [5] => [[5],[]] => 6
0001 => [4,1] => [[4,4],[3]] => 9
0010 => [3,2] => [[4,3],[2]] => 11
0011 => [3,1,1] => [[3,3,3],[2,2]] => 10
0100 => [2,3] => [[4,2],[1]] => 11
0101 => [2,2,1] => [[3,3,2],[2,1]] => 13
0110 => [2,1,2] => [[3,2,2],[1,1]] => 12
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 9
1000 => [1,4] => [[4,1],[]] => 9
1001 => [1,3,1] => [[3,3,1],[2]] => 12
1010 => [1,2,2] => [[3,2,1],[1]] => 13
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 11
1100 => [1,1,3] => [[3,1,1],[]] => 10
1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 11
1110 => [1,1,1,2] => [[2,1,1,1],[]] => 9
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 6
00000 => [6] => [[6],[]] => 7
00001 => [5,1] => [[5,5],[4]] => 11
00010 => [4,2] => [[5,4],[3]] => 14
00011 => [4,1,1] => [[4,4,4],[3,3]] => 13
00100 => [3,3] => [[5,3],[2]] => 15
00101 => [3,2,1] => [[4,4,3],[3,2]] => 18
00110 => [3,1,2] => [[4,3,3],[2,2]] => 17
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => 13
01000 => [2,4] => [[5,2],[1]] => 14
01001 => [2,3,1] => [[4,4,2],[3,1]] => 19
01010 => [2,2,2] => [[4,3,2],[2,1]] => 21
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => 18
01100 => [2,1,3] => [[4,2,2],[1,1]] => 17
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => 19
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => 16
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => 11
10000 => [1,5] => [[5,1],[]] => 11
10001 => [1,4,1] => [[4,4,1],[3]] => 16
10010 => [1,3,2] => [[4,3,1],[2]] => 19
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 17
10100 => [1,2,3] => [[4,2,1],[1]] => 18
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 21
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 19
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 14
11000 => [1,1,4] => [[4,1,1],[]] => 13
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => 17
11010 => [1,1,2,2] => [[3,2,1,1],[1]] => 18
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => 15
11100 => [1,1,1,3] => [[3,1,1,1],[]] => 13
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => 14
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => 11
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => 7
000000 => [7] => [[7],[]] => 8
000001 => [6,1] => [[6,6],[5]] => 13
000010 => [5,2] => [[6,5],[4]] => 17
000011 => [5,1,1] => [[5,5,5],[4,4]] => 16
000100 => [4,3] => [[6,4],[3]] => 19
000101 => [4,2,1] => [[5,5,4],[4,3]] => 23
000110 => [4,1,2] => [[5,4,4],[3,3]] => 22
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => 17
001000 => [3,4] => [[6,3],[2]] => 19
001001 => [3,3,1] => [[5,5,3],[4,2]] => 26
001010 => [3,2,2] => [[5,4,3],[3,2]] => 29
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => 25
001100 => [3,1,3] => [[5,3,3],[2,2]] => 24
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => 27
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => 23
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => 16
010000 => [2,5] => [[6,2],[1]] => 17
010001 => [2,4,1] => [[5,5,2],[4,1]] => 25
010010 => [2,3,2] => [[5,4,2],[3,1]] => 30
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => 27
010100 => [2,2,3] => [[5,3,2],[2,1]] => 29
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => 34
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => 31
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => 23
011000 => [2,1,4] => [[5,2,2],[1,1]] => 22
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => 29
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => 31
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => 26
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => 23
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => 25
011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => 20
011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => 13
100000 => [1,6] => [[6,1],[]] => 13
100001 => [1,5,1] => [[5,5,1],[4]] => 20
100010 => [1,4,2] => [[5,4,1],[3]] => 25
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => 23
100100 => [1,3,3] => [[5,3,1],[2]] => 26
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => 31
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => 29
>>> Load all 127 entries. <<<
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
Description
The total number of rook placements on a Ferrers board.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
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.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!