- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ 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] => [3] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => [4,2,2,1] => [7,2] => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => [4,2,2,1] => [7,2] => 0

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

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

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => [2,2,2,1] => [7] => 0

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => [2,2,2,1] => [7] => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [2,2,2,1] => [7] => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [2,2,2,1] => [7] => 0

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

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

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

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

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => [4,2,1,1] => [6,1,1] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => [4,2,1,1] => [6,1,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => [4,3,2] => [6,3] => 0

[1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => [4,3,2] => [6,3] => 0

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

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

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

[1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => [4,2,2] => [6,2] => 0

[1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => [4,2,2] => [6,2] => 0

[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => [3,2,2] => [6,1] => 0

[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => [3,2,2] => [6,1] => 0

[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => [3,2,2] => [6,1] => 0

>>> Load all 1968 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with St000143The largest repeated part of a partition. and St000149The number of cells of the partition whose leg is zero and arm is odd., see [1].

Map

2-conjugate

Description

Return a partition with the same number of odd parts and number of even parts interchanged with the number of cells with zero leg and odd arm length.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by

$s$ and interchanges the number of parts divisible by

$s$ and the number of cells with zero leg length and arm length congruent to

$s-1$ modulo

$s$ .
In particular, for

$s=1$ the involution is conjugation, hence the name.

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

$(n-1,\ldots,2,1)$ .

Map

first row removal

Description

Removes the first entry of an integer partition