- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000815: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 8

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

[2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [3,1] => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 9

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

[2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [3,2] => 12

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

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

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

[2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [3,3] => 14

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

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

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

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

[2,3,5,1,4,6] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [3,2] => 12

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

[2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [3,2] => 12

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

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

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

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

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

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

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

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

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

[2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2,1]] => [2,2,1] => 8

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search for individual values

searching the database for the individual values of this statistic

Description

The number of semistandard Young tableaux of partition weight of given shape.
The weight of a semistandard Young tableaux is the sequence

$(m_1, m_2,\dots)$ , where

$m_i$ is the number of occurrences of the number

$i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence.
Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.

Map

inner shape

Description

The inner shape of a skew partition.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .