- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤ

Values

[[2,1],[1]] => [1] => 10 => 01 => 1

[[3,1],[1]] => [1] => 10 => 01 => 1

[[2,2],[1]] => [1] => 10 => 01 => 1

[[3,2],[2]] => [2] => 100 => 101 => 2

[[2,2,1],[1,1]] => [1,1] => 110 => 011 => 1

[[2,1,1],[1]] => [1] => 10 => 01 => 1

[[3,2,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,1],[1]] => [1] => 10 => 01 => 1

[[3,2],[1]] => [1] => 10 => 01 => 1

[[4,2],[2]] => [2] => 100 => 101 => 2

[[3,2,1],[1,1]] => [1,1] => 110 => 011 => 1

[[3,1,1],[1]] => [1] => 10 => 01 => 1

[[4,2,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[3,3],[2]] => [2] => 100 => 101 => 2

[[4,3],[3]] => [3] => 1000 => 0101 => 3

[[2,2,1],[1]] => [1] => 10 => 01 => 1

[[3,3,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[3,2,1],[2]] => [2] => 100 => 101 => 2

[[4,3,1],[3,1]] => [3,1] => 10010 => 01101 => 3

[[2,2,2],[1,1]] => [1,1] => 110 => 011 => 1

[[3,3,2],[2,2]] => [2,2] => 1100 => 1011 => 2

[[3,2,2],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,3,2],[3,2]] => [3,2] => 10100 => 01001 => 3

[[2,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 0111 => 1

[[2,2,1,1],[1,1]] => [1,1] => 110 => 011 => 1

[[3,3,2,1],[2,2,1]] => [2,2,1] => 11010 => 10011 => 2

[[2,1,1,1],[1]] => [1] => 10 => 01 => 1

[[3,2,2,1],[2,1,1]] => [2,1,1] => 10110 => 10001 => 2

[[3,2,1,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,3,2,1],[3,2,1]] => [3,2,1] => 101010 => 011001 => 3

[[5,1],[1]] => [1] => 10 => 01 => 1

[[4,2],[1]] => [1] => 10 => 01 => 1

[[5,2],[2]] => [2] => 100 => 101 => 2

[[4,2,1],[1,1]] => [1,1] => 110 => 011 => 1

[[4,1,1],[1]] => [1] => 10 => 01 => 1

[[5,2,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[3,3],[1]] => [1] => 10 => 01 => 1

[[4,3],[2]] => [2] => 100 => 101 => 2

[[5,3],[3]] => [3] => 1000 => 0101 => 3

[[3,3,1],[1,1]] => [1,1] => 110 => 011 => 1

[[3,2,1],[1]] => [1] => 10 => 01 => 1

[[4,3,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,2,1],[2]] => [2] => 100 => 101 => 2

[[5,3,1],[3,1]] => [3,1] => 10010 => 01101 => 3

[[3,2,2],[1,1]] => [1,1] => 110 => 011 => 1

[[4,3,2],[2,2]] => [2,2] => 1100 => 1011 => 2

[[4,2,2],[2,1]] => [2,1] => 1010 => 1001 => 2

[[5,3,2],[3,2]] => [3,2] => 10100 => 01001 => 3

[[3,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 0111 => 1

[[3,2,1,1],[1,1]] => [1,1] => 110 => 011 => 1

[[4,3,2,1],[2,2,1]] => [2,2,1] => 11010 => 10011 => 2

[[3,1,1,1],[1]] => [1] => 10 => 01 => 1

[[4,2,2,1],[2,1,1]] => [2,1,1] => 10110 => 10001 => 2

[[4,2,1,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[5,3,2,1],[3,2,1]] => [3,2,1] => 101010 => 011001 => 3

[[4,4],[3]] => [3] => 1000 => 0101 => 3

[[5,4],[4]] => [4] => 10000 => 10101 => 4

[[3,3,1],[2]] => [2] => 100 => 101 => 2

[[4,4,1],[3,1]] => [3,1] => 10010 => 01101 => 3

[[4,3,1],[3]] => [3] => 1000 => 0101 => 3

[[5,4,1],[4,1]] => [4,1] => 100010 => 100101 => 4

[[2,2,2],[1]] => [1] => 10 => 01 => 1

[[3,3,2],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,4,2],[3,2]] => [3,2] => 10100 => 01001 => 3

[[3,2,2],[2]] => [2] => 100 => 101 => 2

[[4,3,2],[3,1]] => [3,1] => 10010 => 01101 => 3

[[5,4,2],[4,2]] => [4,2] => 100100 => 101101 => 4

[[2,2,1,1],[1]] => [1] => 10 => 01 => 1

[[3,3,2,1],[2,1,1]] => [2,1,1] => 10110 => 10001 => 2

[[3,3,1,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,4,2,1],[3,2,1]] => [3,2,1] => 101010 => 011001 => 3

[[3,2,1,1],[2]] => [2] => 100 => 101 => 2

[[4,3,2,1],[3,1,1]] => [3,1,1] => 100110 => 011101 => 3

[[4,3,1,1],[3,1]] => [3,1] => 10010 => 01101 => 3

[[5,4,2,1],[4,2,1]] => [4,2,1] => 1001010 => 1001101 => 4

[[3,3,3],[2,2]] => [2,2] => 1100 => 1011 => 2

[[4,4,3],[3,3]] => [3,3] => 11000 => 01011 => 3

[[4,3,3],[3,2]] => [3,2] => 10100 => 01001 => 3

[[5,4,3],[4,3]] => [4,3] => 101000 => 101001 => 4

[[2,2,2,1],[1,1]] => [1,1] => 110 => 011 => 1

[[3,3,3,1],[2,2,1]] => [2,2,1] => 11010 => 10011 => 2

[[3,3,2,1],[2,2]] => [2,2] => 1100 => 1011 => 2

[[4,4,3,1],[3,3,1]] => [3,3,1] => 110010 => 011011 => 3

[[3,2,2,1],[2,1]] => [2,1] => 1010 => 1001 => 2

[[4,3,3,1],[3,2,1]] => [3,2,1] => 101010 => 011001 => 3

[[4,3,2,1],[3,2]] => [3,2] => 10100 => 01001 => 3

[[5,4,3,1],[4,3,1]] => [4,3,1] => 1010010 => 1001001 => 4

[[2,2,2,2],[1,1,1]] => [1,1,1] => 1110 => 0111 => 1

[[3,3,3,2],[2,2,2]] => [2,2,2] => 11100 => 10111 => 2

[[3,3,2,2],[2,2,1]] => [2,2,1] => 11010 => 10011 => 2

[[4,4,3,2],[3,3,2]] => [3,3,2] => 110100 => 010011 => 3

[[3,2,2,2],[2,1,1]] => [2,1,1] => 10110 => 10001 => 2

[[4,3,3,2],[3,2,2]] => [3,2,2] => 101100 => 010001 => 3

[[4,3,2,2],[3,2,1]] => [3,2,1] => 101010 => 011001 => 3

[[5,4,3,2],[4,3,2]] => [4,3,2] => 1010100 => 1011001 => 4

[[2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 11110 => 01111 => 1

[[2,2,2,1,1],[1,1,1]] => [1,1,1] => 1110 => 0111 => 1

[[3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => 111010 => 100111 => 2

[[2,2,1,1,1],[1,1]] => [1,1] => 110 => 011 => 1

[[3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => 110110 => 100011 => 2

[[3,3,2,1,1],[2,2,1]] => [2,2,1] => 11010 => 10011 => 2

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search for individual values

searching the database for the individual values of this statistic

Description

The number of changes of a binary word.
This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Map

zeros to flag zeros

Description

Return a binary word of the same length, such that the number of occurrences of

$10$ in the word obtained by prepending the reverse of the complement equals the number of

$0$ s in the original word.
For example, the image of the word

$w=1\dots1$ is

$1\dots1$ , because

$w$ has no zeros, and

$1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor

$10$ .
On the other hand,

$0\dots0$ must be mapped to

$10\dots10$ if the length is even, and

$010\dots10$ if it is odd.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.

Map

inner shape

Description

The inner shape of a skew partition.