- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001051: Set partitions ⟶ ℤ

Values

[1] => [[1]] => [1] => {{1}} => 1
[2] => [[1,2]] => [1,2] => {{1},{2}} => 2
[1,1] => [[1],[2]] => [2,1] => {{1,2}} => 1
[3] => [[1,2,3]] => [1,2,3] => {{1},{2},{3}} => 3
[2,1] => [[1,2],[3]] => [3,1,2] => {{1,2,3}} => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => {{1,3},{2}} => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => {{1},{2},{3},{4}} => 4
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => {{1,2,3,4}} => 1
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => {{1,3},{2,4}} => 2
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => {{1,2,3,4}} => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => {{1,4},{2,3}} => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => 5
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => {{1,2,3,4,5}} => 1
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => {{1,2,3,4,5}} => 1
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => {{1,3,5},{2,4}} => 1
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => {{1,2,3,4,5}} => 1
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => {{1,2,4,5},{3}} => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => {{1,5},{2,4},{3}} => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => 6
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => {{1,2,3,4,5,6}} => 1
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}} => 2
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => {{1,2,3,4,5,6}} => 1
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}} => 3
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => {{1,2,3,4,5,6}} => 1
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => {{1,3,4,6},{2,5}} => 1
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => {{1,5},{2,6},{3},{4}} => 2
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => {{1,2,5,6},{3},{4}} => 1
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => {{1,2,5,6},{3,4}} => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}} => 1
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}} => 7
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7}} => 1
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7}} => 1
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}} => 1
[4,3] => [[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => {{1,2,3,4,5,6,7}} => 1
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [7,5,6,1,2,3,4] => {{1,4,7},{2,5},{3,6}} => 1
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => {{1,4,7},{2,3,5,6}} => 1
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => {{1,3,5,7},{2,4,6}} => 1
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [6,7,4,5,1,2,3] => {{1,2,3,4,5,6,7}} => 1
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}} => 1
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => {{1,3,5,7},{2,6},{4}} => 1
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => {{1,2,3,4,5,6,7}} => 1
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => {{1,2,6,7},{3,4,5}} => 1
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => {{1,2,6,7},{3,5},{4}} => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => {{1,7},{2,6},{3,5},{4}} => 1
[7,1] => [[1,2,3,4,5,6,7],[8]] => [8,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8}} => 1
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [8,7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8}} => 1
[5,3] => [[1,2,3,4,5],[6,7,8]] => [6,7,8,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => 1
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [8,6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}} => 1
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [8,7,6,1,2,3,4,5] => {{1,2,4,5,7,8},{3,6}} => 1
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}} => 4
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [8,5,6,7,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => 1
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [8,7,5,6,1,2,3,4] => {{1,2,3,4,5,6,7,8}} => 1
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [7,8,4,5,6,1,2,3] => {{1,2,3,4,5,6,7,8}} => 1
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [8,7,4,5,6,1,2,3] => {{1,3,4,5,6,8},{2,7}} => 1
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => {{1,7},{2,8},{3,5},{4,6}} => 2
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}} => 1
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => {{1,10},{2,9},{3,8},{4,7},{5,6}} => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [12,11,10,9,8,7,6,5,4,3,2,1] => {{1,12},{2,11},{3,10},{4,9},{5,8},{6,7}} => 1

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

click to show known generating functions

$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + q^{3}$

$F_{4} = 3\ q + q^{2} + q^{4}$

$F_{5} = 6\ q + q^{5}$

$F_{6} = 7\ q + 2\ q^{2} + q^{3} + q^{6}$

$F_{7} = 14\ q + q^{7}$

Description

The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition.
The bijection between set partitions of

$\{1,\dots,n\}$ into

$k$ blocks and trees with

$n+1-k$ leaves is described in Theorem 1 of [1].

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.

Map

to cycle type

Description

Let

$\pi=c_1\dots c_r$ a permutation of size

$n$ decomposed in its cyclic parts. The associated set partition of

$[n]$ then is

$S=S_1\cup\dots\cup S_r$ such that

$S_i$ is the set of integers in the cycle

$c_i$ .
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].