Your data matches 183 different statistics following compositions of up to 3 maps.
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Matching statistic: St000627
(load all 100 compositions to match this statistic)
Mapping
St000627: 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 => 2 = 3 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 3 = 4 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 1 = 2 - 1
111 => 3 = 4 - 1
0000 => 4 = 5 - 1
1111 => 4 = 5 - 1
00000 => 5 = 6 - 1
11111 => 5 = 6 - 1
000000 => 6 = 7 - 1
111111 => 6 = 7 - 1
Description
The exponent of a binary word. This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.
Matching statistic: St000907
(load all 8 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => 2
1 => ([(0,1)],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
11 => ([(0,2),(2,1)],3)
 => 3
000 => ([(0,3),(2,1),(3,2)],4)
 => 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000657
(load all 26 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 2 = 3 - 1
01 => [1,1] => 1 = 2 - 1
10 => [1,1] => 1 = 2 - 1
11 => [2] => 2 = 3 - 1
000 => [3] => 3 = 4 - 1
001 => [2,1] => 1 = 2 - 1
010 => [1,1,1] => 1 = 2 - 1
011 => [1,2] => 1 = 2 - 1
100 => [1,2] => 1 = 2 - 1
101 => [1,1,1] => 1 = 2 - 1
110 => [2,1] => 1 = 2 - 1
111 => [3] => 3 = 4 - 1
0000 => [4] => 4 = 5 - 1
1111 => [4] => 4 = 5 - 1
00000 => [5] => 5 = 6 - 1
11111 => [5] => 5 = 6 - 1
000000 => [6] => 6 = 7 - 1
111111 => [6] => 6 = 7 - 1
Description
The smallest part of an integer composition.
Matching statistic: St000876
(load all 2 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 2 = 3 - 1
01 => 10 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 3 = 4 - 1
001 => 100 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 110 => 1 = 2 - 1
100 => 100 => 1 = 2 - 1
101 => 110 => 1 = 2 - 1
110 => 110 => 1 = 2 - 1
111 => 111 => 3 = 4 - 1
0000 => 0000 => 4 = 5 - 1
1111 => 1111 => 4 = 5 - 1
00000 => 00000 => 5 = 6 - 1
11111 => 11111 => 5 = 6 - 1
000000 => 000000 => 6 = 7 - 1
111111 => 111111 => 6 = 7 - 1
Description
The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St001267
(load all 3 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 2 = 3 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 3 = 4 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 1 = 2 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 1 = 2 - 1
110 => 011 => 1 = 2 - 1
111 => 111 => 3 = 4 - 1
0000 => 0000 => 4 = 5 - 1
1111 => 1111 => 4 = 5 - 1
00000 => 00000 => 5 = 6 - 1
11111 => 11111 => 5 = 6 - 1
000000 => 000000 => 6 = 7 - 1
111111 => 111111 => 6 = 7 - 1
Description
The length of the Lyndon factorization of the binary word. The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.
Matching statistic: St001437
(load all 2 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 2 = 3 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 3 = 4 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 1 = 2 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 1 = 2 - 1
110 => 011 => 1 = 2 - 1
111 => 111 => 3 = 4 - 1
0000 => 0000 => 4 = 5 - 1
1111 => 1111 => 4 = 5 - 1
00000 => 00000 => 5 = 6 - 1
11111 => 11111 => 5 = 6 - 1
000000 => 000000 => 6 = 7 - 1
111111 => 111111 => 6 = 7 - 1
Description
The flex of a binary word. This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St001884
(load all 5 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001884: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 2 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 2 = 3 - 1
01 => 01 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 3 = 4 - 1
001 => 001 => 1 = 2 - 1
010 => 001 => 1 = 2 - 1
011 => 011 => 1 = 2 - 1
100 => 001 => 1 = 2 - 1
101 => 011 => 1 = 2 - 1
110 => 011 => 1 = 2 - 1
111 => 111 => 3 = 4 - 1
0000 => 0000 => 4 = 5 - 1
1111 => 1111 => 4 = 5 - 1
00000 => 00000 => 5 = 6 - 1
11111 => 11111 => 5 = 6 - 1
000000 => 000000 => 6 = 7 - 1
111111 => 111111 => 6 = 7 - 1
Description
The number of borders of a binary word. A border of a binary word $w$ is a word which is both a prefix and a suffix of $w$.
Matching statistic: St000295
(load all 5 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000295: 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 => 1 = 3 - 2
01 => 01 => 0 = 2 - 2
10 => 01 => 0 = 2 - 2
11 => 11 => 1 = 3 - 2
000 => 000 => 2 = 4 - 2
001 => 001 => 0 = 2 - 2
010 => 001 => 0 = 2 - 2
011 => 011 => 0 = 2 - 2
100 => 001 => 0 = 2 - 2
101 => 011 => 0 = 2 - 2
110 => 011 => 0 = 2 - 2
111 => 111 => 2 = 4 - 2
0000 => 0000 => 3 = 5 - 2
1111 => 1111 => 3 = 5 - 2
00000 => 00000 => 4 = 6 - 2
11111 => 11111 => 4 = 6 - 2
000000 => 000000 => 5 = 7 - 2
111111 => 111111 => 5 = 7 - 2
Description
The length of the border of a binary word. The border of a word is the longest word which is both a proper prefix and a proper suffix, including a possible empty border.
Matching statistic: St000723
(load all 2 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 2
1 => ([(0,1)],2)
 => ([],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
11 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 6
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 6
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 7
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 7
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].
Matching statistic: St000776
(load all 3 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000776: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 2
1 => ([(0,1)],2)
 => ([],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
11 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 6
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 6
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 7
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 7
Description
The maximal multiplicity of an eigenvalue in a graph.
The following 173 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001691The number of kings in a graph. St000543The size of the conjugacy class of a binary word. St000553The number of blocks of a graph. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000775The multiplicity of the largest eigenvalue in a graph. 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. St000982The length of the longest constant 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. 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 $g$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001571The Cartan determinant of the integer partition. St000552The number of cut vertices of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000439The position of the first down step of a Dyck path. St000475The number of parts equal to 1 in a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001654The monophonic hull number of a graph. St001933The largest multiplicity of a part in an integer partition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000056The decomposition (or block) number of a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000617The number of global maxima of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000700The protection number of an ordered tree. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001614The cyclic permutation representation number of a skew partition. St000210Minimum over maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000546The number of global descents of a permutation. St000877The depth of the binary word interpreted as a path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001091The number of parts in an integer partition whose next smaller part has the same size. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000315The number of isolated vertices of a graph. St000906The length of the shortest maximal chain in a poset. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000120The number of left tunnels of a Dyck path. St000487The length of the shortest cycle of a permutation. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001130The number of two successive successions in a permutation. St000850The number of 1/2-balanced pairs in a poset. St001354The number of series nodes in the modular decomposition of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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 $eA$. St000264The girth of a graph, which is not a tree. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. 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. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000806The semiperimeter of the associated bargraph. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001644The dimension of a graph. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001877Number of indecomposable injective modules with projective dimension 2. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000699The toughness times the least common multiple of 1,.