- FindStat

Your data matches 187 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001423
(load all 30 compositions to match this statistic)

Mapping

St001423: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0
=> 0
=> 0
=> 0
=> 0
=> 0
=> 1
=> 0
=> 0
=> 0
=> 0
=> 0
=> 0
=> 1
=> 1
=> 1
=> 0
=> 0
=> 0
=> 0
=> 0
=> 1
=> 1
=> 0
=> 0
=> 0
=> 0
=> 0
=> 1
=> 1

Description

The number of distinct cubes in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words

$u$ such that

$uuu$ is a factor of the word.

Matching statistic: St000761
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents in an integer composition. A composition has an ascent, or rise, at position

$i$ if

$a_i < a_{i+1}$ .

Matching statistic: St000760
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.

Matching statistic: St000764
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong records in an integer composition. A strong record is an element

$a_i$ such that

$a_i > a_j$ for all

$j < i$ . In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.

Matching statistic: St000552
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => ([],1)
 => 0
1 => [1] => ([],1)
 => 0
00 => [2] => ([],2)
 => 0
01 => [1,1] => ([(0,1)],2)
 => 0
10 => [1,1] => ([(0,1)],2)
 => 0
11 => [2] => ([],2)
 => 0
000 => [3] => ([],3)
 => 0
001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
011 => [1,2] => ([(1,2)],3)
 => 0
100 => [1,2] => ([(1,2)],3)
 => 0
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
111 => [3] => ([],3)
 => 0
0000 => [4] => ([],4)
 => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
0111 => [1,3] => ([(2,3)],4)
 => 0
1000 => [1,3] => ([(2,3)],4)
 => 0
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
1111 => [4] => ([],4)
 => 0

Description

The number of cut vertices of a graph. A cut vertex is one whose deletion increases the number of connected components.

Matching statistic: St000660
(load all 4 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => 0
1 => [1] => [1,0]
 => 0
00 => [2] => [1,1,0,0]
 => 0
01 => [1,1] => [1,0,1,0]
 => 0
10 => [1,1] => [1,0,1,0]
 => 0
11 => [2] => [1,1,0,0]
 => 0
000 => [3] => [1,1,1,0,0,0]
 => 1
001 => [2,1] => [1,1,0,0,1,0]
 => 0
010 => [1,1,1] => [1,0,1,0,1,0]
 => 0
011 => [1,2] => [1,0,1,1,0,0]
 => 0
100 => [1,2] => [1,0,1,1,0,0]
 => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => 0
110 => [2,1] => [1,1,0,0,1,0]
 => 0
111 => [3] => [1,1,1,0,0,0]
 => 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 1

Description

The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].

Matching statistic: St001022
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => 0
1 => [1] => [1,0]
 => 0
00 => [2] => [1,1,0,0]
 => 0
01 => [1,1] => [1,0,1,0]
 => 0
10 => [1,1] => [1,0,1,0]
 => 0
11 => [2] => [1,1,0,0]
 => 0
000 => [3] => [1,1,1,0,0,0]
 => 0
001 => [2,1] => [1,1,0,0,1,0]
 => 0
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => 0
100 => [1,2] => [1,0,1,1,0,0]
 => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1
110 => [2,1] => [1,1,0,0,1,0]
 => 0
111 => [3] => [1,1,1,0,0,0]
 => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0

Description

Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001186
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => 0
1 => [1] => [1,0]
 => 0
00 => [2] => [1,1,0,0]
 => 0
01 => [1,1] => [1,0,1,0]
 => 0
10 => [1,1] => [1,0,1,0]
 => 0
11 => [2] => [1,1,0,0]
 => 0
000 => [3] => [1,1,1,0,0,0]
 => 0
001 => [2,1] => [1,1,0,0,1,0]
 => 0
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => 0
100 => [1,2] => [1,0,1,1,0,0]
 => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1
110 => [2,1] => [1,1,0,0,1,0]
 => 0
111 => [3] => [1,1,1,0,0,0]
 => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0

Description

Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001266
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001266: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => 0
1 => [1] => [1,0]
 => 0
00 => [2] => [1,1,0,0]
 => 0
01 => [1,1] => [1,0,1,0]
 => 0
10 => [1,1] => [1,0,1,0]
 => 0
11 => [2] => [1,1,0,0]
 => 0
000 => [3] => [1,1,1,0,0,0]
 => 0
001 => [2,1] => [1,1,0,0,1,0]
 => 0
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => 0
100 => [1,2] => [1,0,1,1,0,0]
 => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1
110 => [2,1] => [1,1,0,0,1,0]
 => 0
111 => [3] => [1,1,1,0,0,0]
 => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0

Description

The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra.

Matching statistic: St001414
(load all 19 compositions to match this statistic)

Mapping

Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0 => 0 => 0
=> 1 => 1 => 0
=> 00 => 00 => 0
=> 10 => 01 => 0
=> 10 => 01 => 0
=> 11 => 11 => 0
=> 000 => 000 => 1
=> 100 => 001 => 0
=> 100 => 001 => 0
=> 110 => 011 => 0
=> 100 => 001 => 0
=> 110 => 011 => 0
=> 110 => 011 => 0
=> 111 => 111 => 1
=> 0000 => 0000 => 1
=> 1000 => 0001 => 1
=> 1000 => 0001 => 1
=> 1010 => 0011 => 0
=> 1000 => 0001 => 1
=> 1100 => 0011 => 0
=> 1010 => 0011 => 0
=> 1110 => 0111 => 0
=> 1000 => 0001 => 1
=> 1010 => 0011 => 0
=> 1100 => 0011 => 0
=> 1110 => 0111 => 0
=> 1010 => 0011 => 0
=> 1110 => 0111 => 0
=> 1110 => 0111 => 0
=> 1111 => 1111 => 1

Description

Half the length of the longest odd length palindromic prefix of a binary word. More precisely, this statistic is the largest number

$k$ such that the word has a palindromic prefix of length

$2k+1$ .

The following 177 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001730The number of times the path corresponding to a binary word crosses the base line. St000920The logarithmic height of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St000052The number of valleys of a Dyck path not on the x-axis. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000386The number of factors DDU in a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000549The number of odd partial sums of an integer partition. St000647The number of big descents of a permutation. St000664The number of right ropes of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000921The number of internal inversions of a binary word. St000944The 3-degree of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules 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. 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. 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. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001394The genus of a permutation. St001484The number of singletons of an integer partition. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001728The number of invisible descents of a permutation. St001797The number of overfull subgraphs of a graph. St001871The number of triconnected components of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000159The number of distinct parts of the integer partition. St000201The number of leaf nodes in a binary tree. St000273The domination number of a graph. St000287The number of connected components of a graph. St000390The number of runs of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. 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. St000783The side length of the largest staircase partition fitting into a partition. St000889The number of alternating sign matrices with the same antidiagonal sums. St000897The number of different multiplicities of parts of an integer partition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001202Call 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 special CNakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001432The order dimension of the partition. St001463The number of distinct columns in the nullspace of a graph. St001829The common independence number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000661The number of rises of length 3 of a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000836The number of descents of distance 2 of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000886The number of permutations with the same antidiagonal sums. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000940The number of characters of the symmetric group whose value on the partition is zero. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001586The number of odd parts smaller than the largest even part in 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001095The number of non-isomorphic posets with precisely one further covering relation. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001656The monophonic position number of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St001175The size of a partition minus the hook length of the base cell. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001271The competition number of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000741The Colin de Verdière graph invariant. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000478Another weight of a partition according to Alladi. St000699The toughness times the least common multiple of 1,. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001812The biclique partition number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000137The Grundy value of an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . 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. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001722The number of minimal chains with small intervals between a binary word and the top element. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. 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. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. 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. St001128The exponens consonantiae of a partition. St000782The indicator function of whether a given perfect matching is an L & P matching.