- FindStat

Identifier

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 17 + 6\ q + q^{2}$

$F_{5} = 70 + 41\ q + 8\ q^{2} + q^{3}$

$F_{6} = 349 + 274\ q + 86\ q^{2} + 10\ q^{3} + q^{4}$

$F_{7} = 2017 + 2040\ q + 803\ q^{2} + 167\ q^{3} + 12\ q^{4} + q^{5}$

Description

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .