- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => [3,2] => 10 => 0

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

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [2,2] => 00 => 0

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

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

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [3,1] => 11 => 0

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

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [2,1] => 01 => 0

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

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

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [1,1] => 11 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => [3,3,1,1] => 1111 => 0

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => [4,1,1,1] => 0111 => 0

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

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

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => [3,1,1,1] => 1111 => 0

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

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

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

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

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [1,1,1,1] => 1111 => 0

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

[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [1,1,1,1] => 1111 => 0

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [1,1,1,1] => 1111 => 0

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

[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => [4,3,2] => 010 => 0

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

[1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => [3,3,2] => 110 => 1

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

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

[1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => [4,2,2] => 000 => 0

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

[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => [3,2,2] => 100 => 1

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

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

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search for individual values

searching the database for the individual values of this statistic

Description

The number of internal inversions of a binary word.
Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted.
An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

odd parts

Description

Return the binary word indicating which parts of the partition are odd.

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.