Your data matches 358 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000847
(load all 23 compositions to match this statistic)
Mapping
St000847: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 = 2 - 1
1 => 1 = 2 - 1
00 => 1 = 2 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 1 = 2 - 1
000 => 1 = 2 - 1
001 => 1 = 2 - 1
010 => 2 = 3 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 2 = 3 - 1
110 => 1 = 2 - 1
111 => 1 = 2 - 1
0000 => 1 = 2 - 1
1111 => 1 = 2 - 1
00000 => 1 = 2 - 1
11111 => 1 = 2 - 1
000000 => 1 = 2 - 1
111111 => 1 = 2 - 1
Description
The number of standard Young tableaux whose descent set is the binary word. A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation. For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
Matching statistic: St000921
(load all 25 compositions to match this statistic)
Mapping
Mp00278: Binary words —rowmotion⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 2 - 2
1 => 1 => 0 = 2 - 2
00 => 00 => 0 = 2 - 2
01 => 10 => 0 = 2 - 2
10 => 01 => 0 = 2 - 2
11 => 11 => 0 = 2 - 2
000 => 000 => 0 = 2 - 2
001 => 010 => 0 = 2 - 2
010 => 100 => 1 = 3 - 2
011 => 101 => 0 = 2 - 2
100 => 001 => 0 = 2 - 2
101 => 110 => 1 = 3 - 2
110 => 011 => 0 = 2 - 2
111 => 111 => 0 = 2 - 2
0000 => 0000 => 0 = 2 - 2
1111 => 1111 => 0 = 2 - 2
00000 => 00000 => 0 = 2 - 2
11111 => 11111 => 0 = 2 - 2
000000 => 000000 => 0 = 2 - 2
111111 => 111111 => 0 = 2 - 2
Description
The number of internal inversions of a binary word. Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted. An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
Matching statistic: St001730
(load all 49 compositions to match this statistic)
Mapping
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 2 - 2
1 => 1 => 0 = 2 - 2
00 => 00 => 0 = 2 - 2
01 => 10 => 0 = 2 - 2
10 => 01 => 0 = 2 - 2
11 => 11 => 0 = 2 - 2
000 => 000 => 0 = 2 - 2
001 => 010 => 0 = 2 - 2
010 => 100 => 1 = 3 - 2
011 => 110 => 0 = 2 - 2
100 => 001 => 0 = 2 - 2
101 => 011 => 1 = 3 - 2
110 => 101 => 0 = 2 - 2
111 => 111 => 0 = 2 - 2
0000 => 0000 => 0 = 2 - 2
1111 => 1111 => 0 = 2 - 2
00000 => 00000 => 0 = 2 - 2
11111 => 11111 => 0 = 2 - 2
000000 => 000000 => 0 = 2 - 2
111111 => 111111 => 0 = 2 - 2
Description
The number of times the path corresponding to a binary word crosses the base line. Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001656
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
1 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
11 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
000 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2
Description
The monophonic position number of a graph. A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.
Matching statistic: St000047
(load all 9 compositions to match this statistic)
Mapping
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => 1 = 2 - 1
1 => 1 => [1] => 1 = 2 - 1
00 => 00 => [2] => 1 = 2 - 1
01 => 10 => [1,1] => 1 = 2 - 1
10 => 01 => [1,1] => 1 = 2 - 1
11 => 11 => [2] => 1 = 2 - 1
000 => 000 => [3] => 1 = 2 - 1
001 => 100 => [1,2] => 1 = 2 - 1
010 => 001 => [2,1] => 2 = 3 - 1
011 => 101 => [1,1,1] => 1 = 2 - 1
100 => 010 => [1,1,1] => 1 = 2 - 1
101 => 110 => [2,1] => 2 = 3 - 1
110 => 011 => [1,2] => 1 = 2 - 1
111 => 111 => [3] => 1 = 2 - 1
0000 => 0000 => [4] => 1 = 2 - 1
1111 => 1111 => [4] => 1 = 2 - 1
00000 => 00000 => [5] => 1 = 2 - 1
11111 => 11111 => [5] => 1 = 2 - 1
000000 => 000000 => [6] => 1 = 2 - 1
111111 => 111111 => [6] => 1 = 2 - 1
Description
The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000758
(load all 7 compositions to match this statistic)
Mapping
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => 1 = 2 - 1
1 => 1 => [1] => 1 = 2 - 1
00 => 00 => [2] => 1 = 2 - 1
01 => 10 => [1,1] => 1 = 2 - 1
10 => 01 => [1,1] => 1 = 2 - 1
11 => 11 => [2] => 1 = 2 - 1
000 => 000 => [3] => 1 = 2 - 1
001 => 010 => [1,1,1] => 1 = 2 - 1
010 => 100 => [1,2] => 2 = 3 - 1
011 => 110 => [2,1] => 1 = 2 - 1
100 => 001 => [2,1] => 1 = 2 - 1
101 => 011 => [1,2] => 2 = 3 - 1
110 => 101 => [1,1,1] => 1 = 2 - 1
111 => 111 => [3] => 1 = 2 - 1
0000 => 0000 => [4] => 1 = 2 - 1
1111 => 1111 => [4] => 1 = 2 - 1
00000 => 00000 => [5] => 1 = 2 - 1
11111 => 11111 => [5] => 1 = 2 - 1
000000 => 000000 => [6] => 1 = 2 - 1
111111 => 111111 => [6] => 1 = 2 - 1
Description
The length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000760
(load all 9 compositions to match this statistic)
Mapping
Mp00136: Binary words —rotate back-to-front⟶ Binary words
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
0 => 0 => [1] => 1 = 2 - 1
1 => 1 => [1] => 1 = 2 - 1
00 => 00 => [2] => 1 = 2 - 1
01 => 10 => [1,1] => 1 = 2 - 1
10 => 01 => [1,1] => 1 = 2 - 1
11 => 11 => [2] => 1 = 2 - 1
000 => 000 => [3] => 1 = 2 - 1
001 => 100 => [1,2] => 1 = 2 - 1
010 => 001 => [2,1] => 2 = 3 - 1
011 => 101 => [1,1,1] => 1 = 2 - 1
100 => 010 => [1,1,1] => 1 = 2 - 1
101 => 110 => [2,1] => 2 = 3 - 1
110 => 011 => [1,2] => 1 = 2 - 1
111 => 111 => [3] => 1 = 2 - 1
0000 => 0000 => [4] => 1 = 2 - 1
1111 => 1111 => [4] => 1 = 2 - 1
00000 => 00000 => [5] => 1 = 2 - 1
11111 => 11111 => [5] => 1 = 2 - 1
000000 => 000000 => [6] => 1 = 2 - 1
111111 => 111111 => [6] => 1 = 2 - 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 7 compositions to match this statistic)
Mapping
Mp00135: Binary words —rotate front-to-back⟶ Binary words
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
0 => 0 => [1] => 1 = 2 - 1
1 => 1 => [1] => 1 = 2 - 1
00 => 00 => [2] => 1 = 2 - 1
01 => 10 => [1,1] => 1 = 2 - 1
10 => 01 => [1,1] => 1 = 2 - 1
11 => 11 => [2] => 1 = 2 - 1
000 => 000 => [3] => 1 = 2 - 1
001 => 010 => [1,1,1] => 1 = 2 - 1
010 => 100 => [1,2] => 2 = 3 - 1
011 => 110 => [2,1] => 1 = 2 - 1
100 => 001 => [2,1] => 1 = 2 - 1
101 => 011 => [1,2] => 2 = 3 - 1
110 => 101 => [1,1,1] => 1 = 2 - 1
111 => 111 => [3] => 1 = 2 - 1
0000 => 0000 => [4] => 1 = 2 - 1
1111 => 1111 => [4] => 1 = 2 - 1
00000 => 00000 => [5] => 1 = 2 - 1
11111 => 11111 => [5] => 1 = 2 - 1
000000 => 000000 => [6] => 1 = 2 - 1
111111 => 111111 => [6] => 1 = 2 - 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: St001066
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 1 = 2 - 1
1 => [1] => [1,0]
 => 1 = 2 - 1
00 => [2] => [1,1,0,0]
 => 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
10 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
11 => [2] => [1,1,0,0]
 => 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001238
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001238: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 1 = 2 - 1
1 => [1] => [1,0]
 => 1 = 2 - 1
00 => [2] => [1,1,0,0]
 => 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
10 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
11 => [2] => [1,1,0,0]
 => 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
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 348 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to 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. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-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. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001423The number of distinct cubes in a binary word. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000048The multinomial of the parts of a 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. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. 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. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000783The side length of the largest staircase partition fitting into a partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. 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. 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. St000920The logarithmic height of a Dyck path. St000948The chromatic discriminant of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000959The number of strong Bruhat factorizations of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001313The number of Dyck paths above the lattice path given by a binary word. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001432The order dimension of the partition. 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. St001616The number of neutral elements in a lattice. St001627The number of coloured connected graphs 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. St001679The number of subsets of a lattice whose meet is the bottom element. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001754The number of tolerances of a finite lattice. 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. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. 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. St001917The order of toric promotion on the set of labellings of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000002The number of occurrences of the pattern 123 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000218The number of occurrences of the pattern 213 in a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. 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. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing 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 $3$. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000534The number of 2-rises of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000663The number of right floats of a permutation. St000682The Grundy value of Welter's game on a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000731The number of double exceedences of a permutation. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000877The depth of the binary word interpreted as a path. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001172The number of 1-rises at odd height of a Dyck path. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001911A descent variant minus the number of inversions. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000379The number of Hamiltonian cycles in a graph. St001871The number of triconnected components of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000619The number of cyclic descents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000886The number of permutations with the same antidiagonal sums. St000908The length of the shortest maximal antichain in a poset. St000958The number of Bruhat factorizations of a permutation. 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 $c_{n−1}$. St001052The length of the exterior of a permutation. St001128The exponens consonantiae of a partition. 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. St001220The width of a permutation. St001271The competition number of a graph. St001487The number of inner corners of a skew partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000710The number of big 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. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in 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 $K[x]/(x^n)$. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001305The number of induced cycles on four vertices in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001552The number of inversions between excedances and fixed points of a permutation. St001731The factorization defect of a permutation. St001856The number of edges in the reduced word graph of a permutation. St000570The Edelman-Greene number of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000741The Colin de Verdière graph invariant. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001282The number of graphs with the same chromatic polynomial. St001498The normalised height of a Nakayama algebra with magnitude 1. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000219The number of occurrences of the pattern 231 in a permutation. St000699The toughness times the least common multiple of 1,. St000260The radius of a connected graph. St001964The interval resolution global dimension of a poset. St001625The Möbius invariant of a lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000806The semiperimeter of the associated bargraph. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001592The maximal number of simple paths between any two different vertices of a graph. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001812The biclique partition number of a graph. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001722The number of minimal chains with small intervals between a binary word and the top element. St000941The number of characters of the symmetric group whose value on the partition is even. St001060The distinguishing index of a graph. St000464The Schultz index of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001545The second Elser number of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral.