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Your data matches 197 different statistics following compositions of up to 3 maps.
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Matching statistic: St000904
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 2
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 3
[1,1,2,1] => 3
[1,1,3] => 2
[1,2,1,1] => 3
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 1
[2,1,1,1] => 3
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 1
[3,1,1] => 2
[3,2] => 1
[4,1] => 1
[5] => 1
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001933
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 2
[2] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 3
[1,2] => [2,1]
=> 1
[2,1] => [2,1]
=> 1
[3] => [3]
=> 1
[1,1,1,1] => [1,1,1,1]
=> 4
[1,1,2] => [2,1,1]
=> 2
[1,2,1] => [2,1,1]
=> 2
[1,3] => [3,1]
=> 1
[2,1,1] => [2,1,1]
=> 2
[2,2] => [2,2]
=> 2
[3,1] => [3,1]
=> 1
[4] => [4]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,1,1,2] => [2,1,1,1]
=> 3
[1,1,2,1] => [2,1,1,1]
=> 3
[1,1,3] => [3,1,1]
=> 2
[1,2,1,1] => [2,1,1,1]
=> 3
[1,2,2] => [2,2,1]
=> 2
[1,3,1] => [3,1,1]
=> 2
[1,4] => [4,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> 3
[2,1,2] => [2,2,1]
=> 2
[2,2,1] => [2,2,1]
=> 2
[2,3] => [3,2]
=> 1
[3,1,1] => [3,1,1]
=> 2
[3,2] => [3,2]
=> 1
[4,1] => [4,1]
=> 1
[5] => [5]
=> 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001091
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 1 - 1
[1,1] => [1,1]
=> 1 = 2 - 1
[2] => [2]
=> 0 = 1 - 1
[1,1,1] => [1,1,1]
=> 2 = 3 - 1
[1,2] => [2,1]
=> 0 = 1 - 1
[2,1] => [2,1]
=> 0 = 1 - 1
[3] => [3]
=> 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> 3 = 4 - 1
[1,1,2] => [2,1,1]
=> 1 = 2 - 1
[1,2,1] => [2,1,1]
=> 1 = 2 - 1
[1,3] => [3,1]
=> 0 = 1 - 1
[2,1,1] => [2,1,1]
=> 1 = 2 - 1
[2,2] => [2,2]
=> 1 = 2 - 1
[3,1] => [3,1]
=> 0 = 1 - 1
[4] => [4]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 4 = 5 - 1
[1,1,1,2] => [2,1,1,1]
=> 2 = 3 - 1
[1,1,2,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,1,3] => [3,1,1]
=> 1 = 2 - 1
[1,2,1,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,2] => [2,2,1]
=> 1 = 2 - 1
[1,3,1] => [3,1,1]
=> 1 = 2 - 1
[1,4] => [4,1]
=> 0 = 1 - 1
[2,1,1,1] => [2,1,1,1]
=> 2 = 3 - 1
[2,1,2] => [2,2,1]
=> 1 = 2 - 1
[2,2,1] => [2,2,1]
=> 1 = 2 - 1
[2,3] => [3,2]
=> 0 = 1 - 1
[3,1,1] => [3,1,1]
=> 1 = 2 - 1
[3,2] => [3,2]
=> 0 = 1 - 1
[4,1] => [4,1]
=> 0 = 1 - 1
[5] => [5]
=> 0 = 1 - 1
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1
[1,1] => [1,1]
=> 110 => 2
[2] => [2]
=> 100 => 1
[1,1,1] => [1,1,1]
=> 1110 => 3
[1,2] => [2,1]
=> 1010 => 1
[2,1] => [2,1]
=> 1010 => 1
[3] => [3]
=> 1000 => 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 4
[1,1,2] => [2,1,1]
=> 10110 => 2
[1,2,1] => [2,1,1]
=> 10110 => 2
[1,3] => [3,1]
=> 10010 => 1
[2,1,1] => [2,1,1]
=> 10110 => 2
[2,2] => [2,2]
=> 1100 => 2
[3,1] => [3,1]
=> 10010 => 1
[4] => [4]
=> 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 5
[1,1,1,2] => [2,1,1,1]
=> 101110 => 3
[1,1,2,1] => [2,1,1,1]
=> 101110 => 3
[1,1,3] => [3,1,1]
=> 100110 => 2
[1,2,1,1] => [2,1,1,1]
=> 101110 => 3
[1,2,2] => [2,2,1]
=> 11010 => 2
[1,3,1] => [3,1,1]
=> 100110 => 2
[1,4] => [4,1]
=> 100010 => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => 3
[2,1,2] => [2,2,1]
=> 11010 => 2
[2,2,1] => [2,2,1]
=> 11010 => 2
[2,3] => [3,2]
=> 10100 => 1
[3,1,1] => [3,1,1]
=> 100110 => 2
[3,2] => [3,2]
=> 10100 => 1
[4,1] => [4,1]
=> 100010 => 1
[5] => [5]
=> 100000 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1
[1,1] => [1,1]
=> 110 => 2
[2] => [2]
=> 100 => 1
[1,1,1] => [1,1,1]
=> 1110 => 3
[1,2] => [2,1]
=> 1010 => 1
[2,1] => [2,1]
=> 1010 => 1
[3] => [3]
=> 1000 => 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 4
[1,1,2] => [2,1,1]
=> 10110 => 2
[1,2,1] => [2,1,1]
=> 10110 => 2
[1,3] => [3,1]
=> 10010 => 1
[2,1,1] => [2,1,1]
=> 10110 => 2
[2,2] => [2,2]
=> 1100 => 2
[3,1] => [3,1]
=> 10010 => 1
[4] => [4]
=> 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 5
[1,1,1,2] => [2,1,1,1]
=> 101110 => 3
[1,1,2,1] => [2,1,1,1]
=> 101110 => 3
[1,1,3] => [3,1,1]
=> 100110 => 2
[1,2,1,1] => [2,1,1,1]
=> 101110 => 3
[1,2,2] => [2,2,1]
=> 11010 => 2
[1,3,1] => [3,1,1]
=> 100110 => 2
[1,4] => [4,1]
=> 100010 => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => 3
[2,1,2] => [2,2,1]
=> 11010 => 2
[2,2,1] => [2,2,1]
=> 11010 => 2
[2,3] => [3,2]
=> 10100 => 1
[3,1,1] => [3,1,1]
=> 100110 => 2
[3,2] => [3,2]
=> 10100 => 1
[4,1] => [4,1]
=> 100010 => 1
[5] => [5]
=> 100000 => 1
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001515
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> 2
[2] => [1] => [1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,2] => [1,1] => [1,0,1,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> 1
[3] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,3] => [1,1] => [1,0,1,0]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,2] => [2] => [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> 1
[4] => [1] => [1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,4] => [1,1] => [1,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3] => [1,1] => [1,0,1,0]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 1
[4,1] => [1,1] => [1,0,1,0]
=> 1
[5] => [1] => [1,0]
=> 1
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St001571
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1
[1,1] => [1,1]
=> [2]
=> 2
[2] => [2]
=> [1,1]
=> 1
[1,1,1] => [1,1,1]
=> [3]
=> 3
[1,2] => [2,1]
=> [2,1]
=> 1
[2,1] => [2,1]
=> [2,1]
=> 1
[3] => [3]
=> [1,1,1]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 4
[1,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,2,1] => [2,1,1]
=> [3,1]
=> 2
[1,3] => [3,1]
=> [2,1,1]
=> 1
[2,1,1] => [2,1,1]
=> [3,1]
=> 2
[2,2] => [2,2]
=> [2,2]
=> 2
[3,1] => [3,1]
=> [2,1,1]
=> 1
[4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 5
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 3
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 3
[1,2,2] => [2,2,1]
=> [3,2]
=> 2
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 2
[1,4] => [4,1]
=> [2,1,1,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 3
[2,1,2] => [2,2,1]
=> [3,2]
=> 2
[2,2,1] => [2,2,1]
=> [3,2]
=> 2
[2,3] => [3,2]
=> [2,2,1]
=> 1
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 2
[3,2] => [3,2]
=> [2,2,1]
=> 1
[4,1] => [4,1]
=> [2,1,1,1]
=> 1
[5] => [5]
=> [1,1,1,1,1]
=> 1
Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001180
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001180: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001180: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2 = 1 + 1
[1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[2] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[2,2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[3,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[5] => [1] => [1,0]
=> 2 = 1 + 1
Description
Number of indecomposable injective modules with projective dimension at most 1.
Matching statistic: St001211
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2 = 1 + 1
[1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[2] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[2,2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[3,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[5] => [1] => [1,0]
=> 2 = 1 + 1
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
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Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2 = 1 + 1
[1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[2] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[2,2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[3,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[5] => [1] => [1,0]
=> 2 = 1 + 1
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
The following 187 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000899The maximal number of repetitions of an integer composition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001399The distinguishing number of a poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001733The number of weak left to right maxima of a Dyck path. St001884The number of borders of a binary word. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000144The pyramid weight of the Dyck path. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000356The number of occurrences of the pattern 13-2. St000672The number of minimal elements in Bruhat order not less than the permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001777The number of weak descents in an integer composition. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000420The number of Dyck paths that are weakly above a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001597The Frobenius rank of a skew partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000668The least common multiple of the parts of the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000680The Grundy value for Hackendot on posets. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000815The number of semistandard Young tableaux of partition weight of given shape. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000906The length of the shortest maximal chain in a poset. St000933The number of multipartitions of sizes given by an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001118The acyclic chromatic index of a graph. St000850The number of 1/2-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St001330The hat guessing number of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St000650The number of 3-rises of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The 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 $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000454The largest eigenvalue of a graph if it is integral. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000075The orbit size of a standard tableau under promotion. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation.
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