- FindStat

Identifier

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤ (values match St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.)

Values

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

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

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

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

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

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

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

[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0] => 2

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

[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0] => 2

[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0] => 2

[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0] => 2

[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2

[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2

[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2

[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2

[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2

[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,5,2,3,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,5,2,4,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,5,3,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,5,3,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,5,4,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[1,5,4,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[2,1,3,4,5] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 2

[2,1,3,5,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2

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

[2,1,4,5,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2

[2,1,5,3,4] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2

[2,1,5,4,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2

[2,3,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,3,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

[2,3,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[2,4,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[2,4,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[3,1,2,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,1,2,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

[3,1,4,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[3,2,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,2,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

[3,2,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[3,4,1,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[3,4,2,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[4,1,2,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[4,1,3,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[4,2,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[4,2,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[4,3,1,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[4,3,2,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 2

[1,2,3,4,6,5] => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2

[1,2,3,5,4,6] => 11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2

[1,2,3,5,6,4] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,2,3,6,4,5] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,2,3,6,5,4] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,2,4,3,5,6] => 11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

[1,2,4,3,6,5] => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 2

[1,2,4,5,3,6] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2

[1,2,4,5,6,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,4,6,3,5] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,4,6,5,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,5,3,4,6] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2

[1,2,5,3,6,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,5,4,3,6] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2

[1,2,5,4,6,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,5,6,3,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,5,6,4,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,6,3,4,5] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,6,3,5,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,6,4,3,5] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,6,4,5,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,6,5,3,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,2,6,5,4,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,3,2,4,5,6] => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2

[1,3,2,4,6,5] => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2

[1,3,2,5,4,6] => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2

[1,3,2,5,6,4] => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2

[1,3,2,6,4,5] => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2

[1,3,2,6,5,4] => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2

[1,3,4,2,5,6] => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2

[1,3,4,2,6,5] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2

[1,3,4,5,2,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2

[1,3,4,5,6,2] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,3,4,6,2,5] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,3,4,6,5,2] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,3,5,2,4,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2

[1,3,5,2,6,4] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,3,5,4,2,6] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2

[1,3,5,4,6,2] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,3,5,6,2,4] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,3,5,6,4,2] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

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searching the database for the individual values of this statistic

Description

The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.