Your data matches 546 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000768
(load all 4 compositions to match this statistic)
Mapping
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,1] => 0 = 1 - 1
[2] => 0 = 1 - 1
[1,1,1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[3] => 0 = 1 - 1
[1,1,1,1] => 0 = 1 - 1
[1,1,2] => 0 = 1 - 1
[1,2,1] => 1 = 2 - 1
[1,3] => 0 = 1 - 1
[2,1,1] => 0 = 1 - 1
[2,2] => 0 = 1 - 1
[3,1] => 0 = 1 - 1
[4] => 0 = 1 - 1
[1,1,1,1,1] => 0 = 1 - 1
[1,1,1,2] => 0 = 1 - 1
[1,1,2,1] => 1 = 2 - 1
[1,1,3] => 0 = 1 - 1
[1,2,1,1] => 1 = 2 - 1
[1,2,2] => 0 = 1 - 1
[1,3,1] => 1 = 2 - 1
[1,4] => 0 = 1 - 1
[2,1,1,1] => 0 = 1 - 1
[2,1,2] => 0 = 1 - 1
[2,2,1] => 0 = 1 - 1
[2,3] => 0 = 1 - 1
[3,1,1] => 0 = 1 - 1
[3,2] => 0 = 1 - 1
[4,1] => 0 = 1 - 1
[5] => 0 = 1 - 1
Description
The number of peaks in an integer composition. A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Matching statistic: St001025
(load all 33 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 0 = 1 - 1
[1,1] => [1,0,1,0]
 => 0 = 1 - 1
[2] => [1,1,0,0]
 => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
 => 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
 => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
Description
Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000183
(load all 6 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000183: 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]
 => [1,1]
 => 1
[2] => [2]
 => [2]
 => 1
[1,1,1] => [1,1,1]
 => [1,1,1]
 => 1
[1,2] => [2,1]
 => [3]
 => 1
[2,1] => [2,1]
 => [3]
 => 1
[3] => [3]
 => [2,1]
 => 1
[1,1,1,1] => [1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,2] => [2,1,1]
 => [3,1]
 => 1
[1,2,1] => [2,1,1]
 => [3,1]
 => 1
[1,3] => [3,1]
 => [2,1,1]
 => 1
[2,1,1] => [2,1,1]
 => [3,1]
 => 1
[2,2] => [2,2]
 => [4]
 => 1
[3,1] => [3,1]
 => [2,1,1]
 => 1
[4] => [4]
 => [2,2]
 => 2
[1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,1,3] => [3,1,1]
 => [2,1,1,1]
 => 1
[1,2,1,1] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,2] => [2,2,1]
 => [5]
 => 1
[1,3,1] => [3,1,1]
 => [2,1,1,1]
 => 1
[1,4] => [4,1]
 => [3,2]
 => 2
[2,1,1,1] => [2,1,1,1]
 => [3,1,1]
 => 1
[2,1,2] => [2,2,1]
 => [5]
 => 1
[2,2,1] => [2,2,1]
 => [5]
 => 1
[2,3] => [3,2]
 => [4,1]
 => 1
[3,1,1] => [3,1,1]
 => [2,1,1,1]
 => 1
[3,2] => [3,2]
 => [4,1]
 => 1
[4,1] => [4,1]
 => [3,2]
 => 2
[5] => [5]
 => [2,2,1]
 => 2
Description
The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.
Matching statistic: St000628
(load all 5 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000628: 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 => 1
[2] => [2]
 => 100 => 1
[1,1,1] => [1,1,1]
 => 1110 => 1
[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 => 1
[1,1,2] => [2,1,1]
 => 10110 => 1
[1,2,1] => [2,1,1]
 => 10110 => 1
[1,3] => [3,1]
 => 10010 => 1
[2,1,1] => [2,1,1]
 => 10110 => 1
[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 => 1
[1,1,1,2] => [2,1,1,1]
 => 101110 => 1
[1,1,2,1] => [2,1,1,1]
 => 101110 => 1
[1,1,3] => [3,1,1]
 => 100110 => 2
[1,2,1,1] => [2,1,1,1]
 => 101110 => 1
[1,2,2] => [2,2,1]
 => 11010 => 1
[1,3,1] => [3,1,1]
 => 100110 => 2
[1,4] => [4,1]
 => 100010 => 1
[2,1,1,1] => [2,1,1,1]
 => 101110 => 1
[2,1,2] => [2,2,1]
 => 11010 => 1
[2,2,1] => [2,2,1]
 => 11010 => 1
[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 balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000785
(load all 5 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000785: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,1] => ([(0,1)],2)
 => ([],1)
 => 1
[2] => ([],2)
 => ([],2)
 => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,2] => ([(1,2)],3)
 => ([],2)
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3] => ([],3)
 => ([],3)
 => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[4] => ([],4)
 => ([],4)
 => 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4] => ([(3,4)],5)
 => ([],4)
 => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5] => ([],5)
 => ([],5)
 => 1
Description
The number of distinct colouring schemes of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.
Matching statistic: St000897
(load all 5 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => [1]
 => 1
[1,1] => ([(0,1)],2)
 => [2]
 => 1
[2] => ([],2)
 => [1,1]
 => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
[1,2] => ([(1,2)],3)
 => [2,1]
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
[3] => ([],3)
 => [1,1,1]
 => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
[4] => ([],4)
 => [1,1,1,1]
 => 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
[5] => ([],5)
 => [1,1,1,1,1]
 => 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000965
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000965: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
Description
The sum of the dimension of Ext^i(D(A),A) for i=1,...,g when g denotes the global dimension of the corresponding LNakayama algebra.
Matching statistic: St001066
(load all 12 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001208
(load all 32 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => 1
[1,1] => [1,0,1,0]
 => [2,1] => 1
[2] => [1,1,0,0]
 => [1,2] => 1
[1,1,1] => [1,0,1,0,1,0]
 => [2,1,3] => 1
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 1
Description
The 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 $K[x]/(x^n)$.
Matching statistic: St001238
(load all 10 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001238: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
Description
The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S.
The following 536 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000552The number of cut vertices of a graph. St000664The number of right ropes of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001513The number of nested exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001797The number of overfull subgraphs of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000570The Edelman-Greene number of a permutation. St000657The smallest part of an integer composition. St000701The protection number of a binary tree. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000805The number of peaks of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St000968We 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 $c_{n−1}$. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001597The Frobenius rank of a skew partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000057The Shynar inversion number of a standard tableau. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000317The cycle descent number of a permutation. St000347The inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000389The number of runs of ones of odd length in a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000549The number of odd partial sums of an integer partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000647The number of big descents of a permutation. St000650The number of 3-rises of a permutation. St000658The number of rises of length 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000731The number of double exceedences of a permutation. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000807The sum of the heights of the valleys of the associated bargraph. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001323The independence gap of a graph. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001689The number of celebrities in a graph. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001931The weak major index of an integer composition regarded as a word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000659The number of rises of length at least 2 of a Dyck path. St000872The number of very big descents of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000886The number of permutations with the same antidiagonal sums. St001162The minimum jump of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000649The number of 3-excedences of a permutation. St000682The Grundy value of Welter's game on a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000836The number of descents of distance 2 of a permutation. St000842The breadth of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001731The factorization defect of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000407The number of occurrences of the pattern 2143 in a permutation. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001432The order dimension of the 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000182The number of permutations whose cycle type is the given integer partition. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer 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. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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 $eA$. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001595The number of standard Young tableaux of the skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. 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. 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. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. St001389The number of partitions of the same length below the given integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001322The size of a minimal independent dominating set in a graph. St001490The number of connected components of a skew partition. St000993The multiplicity of the largest part of an integer partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St001363The Euler characteristic of a graph according to Knill. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000287The number of connected components of a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001339The irredundance number of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001890The maximum magnitude of the Möbius function of a poset. St000286The number of connected components of the complement of a graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001964The interval resolution global dimension of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000629The defect of a binary word. St000068The number of minimal elements in a poset. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000079The number of alternating sign matrices for a given Dyck path. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000260The radius of a connected graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000517The Kreweras number of an integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000913The number of ways to refine the partition into singletons. St000933The number of multipartitions of sizes given by an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001924The number of cells in an integer partition whose arm and leg length coincide. St000902 The minimal number of repetitions of an integer composition. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation. St001715The number of non-records in a permutation. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000327The number of cover relations in a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000782The indicator function of whether a given perfect matching is an L & P matching. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001118The acyclic chromatic index of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000022The number of fixed points of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000456The monochromatic index of a connected graph. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000864The number of circled entries of the shifted recording tableau of a permutation. St000929The constant term of the character polynomial of an integer partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000058The order of a permutation. St000084The number of subtrees. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001705The number of occurrences of the pattern 2413 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. 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. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. 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. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000219The number of occurrences of the pattern 231 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000567The sum of the products of all pairs of parts. 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. 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. 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. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.