Identifier
-
Mp00128:
Set partitions
—to composition⟶
Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
{{1}} => [1] => [1] => [1,0] => 0
{{1,2}} => [2] => [2] => [1,1,0,0] => 0
{{1},{2}} => [1,1] => [1,1] => [1,0,1,0] => 1
{{1,2,3}} => [3] => [3] => [1,1,1,0,0,0] => 0
{{1,2},{3}} => [2,1] => [2,1] => [1,1,0,0,1,0] => 1
{{1,3},{2}} => [2,1] => [2,1] => [1,1,0,0,1,0] => 1
{{1},{2,3}} => [1,2] => [1,2] => [1,0,1,1,0,0] => 2
{{1,2,3,4}} => [4] => [4] => [1,1,1,1,0,0,0,0] => 0
{{1,2,3},{4}} => [3,1] => [3,1] => [1,1,1,0,0,0,1,0] => 1
{{1,2,4},{3}} => [3,1] => [3,1] => [1,1,1,0,0,0,1,0] => 1
{{1,2},{3,4}} => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2
{{1,2},{3},{4}} => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
{{1,3,4},{2}} => [3,1] => [3,1] => [1,1,1,0,0,0,1,0] => 1
{{1,3},{2,4}} => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2
{{1,3},{2},{4}} => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
{{1,4},{2,3}} => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2
{{1},{2,3,4}} => [1,3] => [1,3] => [1,0,1,1,1,0,0,0] => 3
{{1,4},{2},{3}} => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
{{1,2,3,4,5}} => [5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,3,4},{5}} => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
{{1,2,3,5},{4}} => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
{{1,2,3},{4,5}} => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,2,3},{4},{5}} => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1,2,4,5},{3}} => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
{{1,2,4},{3,5}} => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,2,4},{3},{5}} => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1,2,5},{3,4}} => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,2},{3,4,5}} => [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
{{1,2},{3,4},{5}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,2,5},{3},{4}} => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1,2},{3,5},{4}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,3,4,5},{2}} => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
{{1,3,4},{2,5}} => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,3,4},{2},{5}} => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1,3,5},{2,4}} => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,3},{2,4,5}} => [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
{{1,3},{2,4},{5}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,3,5},{2},{4}} => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1,3},{2,5},{4}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,4,5},{2,3}} => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,4},{2,3,5}} => [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
{{1,4},{2,3},{5}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,5},{2,3,4}} => [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
{{1},{2,3,4,5}} => [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
{{1,5},{2,3},{4}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1},{2,3},{4,5}} => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
{{1,4,5},{2},{3}} => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1,4},{2,5},{3}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,5},{2,4},{3}} => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1},{2,4},{3,5}} => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
{{1},{2,5},{3,4}} => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
{{1,2,3,4,5,6}} => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,3,4,5},{6}} => [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
{{1,2,3,4,6},{5}} => [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
{{1,2,3,4},{5,6}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,3,4},{5},{6}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,2,3,5,6},{4}} => [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
{{1,2,3,5},{4,6}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,3,5},{4},{6}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,2,3,6},{4,5}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,3},{4,5,6}} => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,2,3},{4,5},{6}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,3,6},{4},{5}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,2,3},{4,6},{5}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,3},{4},{5,6}} => [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1,2,4,5,6},{3}} => [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
{{1,2,4,5},{3,6}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,4,5},{3},{6}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,2,4,6},{3,5}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,4},{3,5,6}} => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,2,4},{3,5},{6}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,4,6},{3},{5}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,2,4},{3,6},{5}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,4},{3},{5,6}} => [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1,2,5,6},{3,4}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,5},{3,4,6}} => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,2,5},{3,4},{6}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,6},{3,4,5}} => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,2},{3,4,5,6}} => [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
{{1,2},{3,4,5},{6}} => [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
{{1,2,6},{3,4},{5}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2},{3,4,6},{5}} => [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
{{1,2},{3,4},{5,6}} => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
{{1,2},{3,4},{5},{6}} => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
{{1,2,5,6},{3},{4}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,2,5},{3,6},{4}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,5},{3},{4,6}} => [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1,2,6},{3,5},{4}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2},{3,5,6},{4}} => [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
{{1,2},{3,5},{4,6}} => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
{{1,2},{3,5},{4},{6}} => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
{{1,2,6},{3},{4,5}} => [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
{{1,2},{3,6},{4,5}} => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
{{1,2},{3,6},{4},{5}} => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
{{1,3,4,5,6},{2}} => [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
{{1,3,4,5},{2,6}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,3,4,5},{2},{6}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
{{1,3,4,6},{2,5}} => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,3,4},{2,5,6}} => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
{{1,3,4},{2,5},{6}} => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,3,4,6},{2},{5}} => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
Foata bijection
Description
The Foata bijection for compositions.
The Foata bijection $\phi$ is a bijection on the set of words whose letters are positive integers. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..
The Foata bijection $\phi$ is a bijection on the set of words whose letters are positive integers. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
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