Your data matches 225 different statistics following compositions of up to 3 maps.
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St000847: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 1
01 => 1
10 => 1
11 => 1
000 => 1
001 => 1
010 => 2
011 => 1
100 => 1
101 => 2
110 => 1
111 => 1
0000 => 1
0001 => 1
0010 => 2
0011 => 1
0100 => 2
0101 => 3
0110 => 2
0111 => 1
1000 => 1
1001 => 2
1010 => 3
1011 => 2
1100 => 1
1101 => 2
1110 => 1
1111 => 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 $010$.
Mp00178: Binary words to compositionInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
St000820: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1] => 1
1 => [1,1] => [2] => 1
00 => [3] => [1] => 1
01 => [2,1] => [1,1] => 1
10 => [1,2] => [1,1] => 1
11 => [1,1,1] => [3] => 1
000 => [4] => [1] => 1
001 => [3,1] => [1,1] => 1
010 => [2,2] => [2] => 1
011 => [2,1,1] => [1,2] => 2
100 => [1,3] => [1,1] => 1
101 => [1,2,1] => [1,1,1] => 1
110 => [1,1,2] => [2,1] => 2
111 => [1,1,1,1] => [4] => 1
0000 => [5] => [1] => 1
0001 => [4,1] => [1,1] => 1
0010 => [3,2] => [1,1] => 1
0011 => [3,1,1] => [1,2] => 2
0100 => [2,3] => [1,1] => 1
0101 => [2,2,1] => [2,1] => 2
0110 => [2,1,2] => [1,1,1] => 1
0111 => [2,1,1,1] => [1,3] => 2
1000 => [1,4] => [1,1] => 1
1001 => [1,3,1] => [1,1,1] => 1
1010 => [1,2,2] => [1,2] => 2
1011 => [1,2,1,1] => [1,1,2] => 3
1100 => [1,1,3] => [2,1] => 2
1101 => [1,1,2,1] => [2,1,1] => 3
1110 => [1,1,1,2] => [3,1] => 2
1111 => [1,1,1,1,1] => [5] => 1
Description
The number of compositions obtained by rotating the composition.
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001774: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> 1
1 => [1] => ([],1)
=> 1
00 => [2] => ([],2)
=> 1
01 => [1,1] => ([(0,1)],2)
=> 1
10 => [1,1] => ([(0,1)],2)
=> 1
11 => [2] => ([],2)
=> 1
000 => [3] => ([],3)
=> 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
011 => [1,2] => ([(1,2)],3)
=> 1
100 => [1,2] => ([(1,2)],3)
=> 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
111 => [3] => ([],3)
=> 1
0000 => [4] => ([],4)
=> 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
0111 => [1,3] => ([(2,3)],4)
=> 1
1000 => [1,3] => ([(2,3)],4)
=> 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
1111 => [4] => ([],4)
=> 1
Description
The degree of the minimal polynomial of the smallest eigenvalue of a graph.
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001775: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> 1
1 => [1] => ([],1)
=> 1
00 => [2] => ([],2)
=> 1
01 => [1,1] => ([(0,1)],2)
=> 1
10 => [1,1] => ([(0,1)],2)
=> 1
11 => [2] => ([],2)
=> 1
000 => [3] => ([],3)
=> 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
011 => [1,2] => ([(1,2)],3)
=> 1
100 => [1,2] => ([(1,2)],3)
=> 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
111 => [3] => ([],3)
=> 1
0000 => [4] => ([],4)
=> 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
0111 => [1,3] => ([(2,3)],4)
=> 1
1000 => [1,3] => ([(2,3)],4)
=> 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
1111 => [4] => ([],4)
=> 1
Description
The degree of the minimal polynomial of the largest eigenvalue of a graph.
Mp00178: Binary words to compositionInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
St000089: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1] => 0 = 1 - 1
1 => [1,1] => [2] => 0 = 1 - 1
00 => [3] => [1] => 0 = 1 - 1
01 => [2,1] => [1,1] => 0 = 1 - 1
10 => [1,2] => [1,1] => 0 = 1 - 1
11 => [1,1,1] => [3] => 0 = 1 - 1
000 => [4] => [1] => 0 = 1 - 1
001 => [3,1] => [1,1] => 0 = 1 - 1
010 => [2,2] => [2] => 0 = 1 - 1
011 => [2,1,1] => [1,2] => 1 = 2 - 1
100 => [1,3] => [1,1] => 0 = 1 - 1
101 => [1,2,1] => [1,1,1] => 0 = 1 - 1
110 => [1,1,2] => [2,1] => 1 = 2 - 1
111 => [1,1,1,1] => [4] => 0 = 1 - 1
0000 => [5] => [1] => 0 = 1 - 1
0001 => [4,1] => [1,1] => 0 = 1 - 1
0010 => [3,2] => [1,1] => 0 = 1 - 1
0011 => [3,1,1] => [1,2] => 1 = 2 - 1
0100 => [2,3] => [1,1] => 0 = 1 - 1
0101 => [2,2,1] => [2,1] => 1 = 2 - 1
0110 => [2,1,2] => [1,1,1] => 0 = 1 - 1
0111 => [2,1,1,1] => [1,3] => 2 = 3 - 1
1000 => [1,4] => [1,1] => 0 = 1 - 1
1001 => [1,3,1] => [1,1,1] => 0 = 1 - 1
1010 => [1,2,2] => [1,2] => 1 = 2 - 1
1011 => [1,2,1,1] => [1,1,2] => 1 = 2 - 1
1100 => [1,1,3] => [2,1] => 1 = 2 - 1
1101 => [1,1,2,1] => [2,1,1] => 1 = 2 - 1
1110 => [1,1,1,2] => [3,1] => 2 = 3 - 1
1111 => [1,1,1,1,1] => [5] => 0 = 1 - 1
Description
The absolute variation of a composition.
Matching statistic: St000769
Mp00234: Binary words valleys-to-peaksBinary words
Mp00097: Binary words delta morphismInteger compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => 0 = 1 - 1
1 => 1 => [1] => 0 = 1 - 1
00 => 01 => [1,1] => 0 = 1 - 1
01 => 10 => [1,1] => 0 = 1 - 1
10 => 11 => [2] => 0 = 1 - 1
11 => 11 => [2] => 0 = 1 - 1
000 => 001 => [2,1] => 1 = 2 - 1
001 => 010 => [1,1,1] => 0 = 1 - 1
010 => 101 => [1,1,1] => 0 = 1 - 1
011 => 101 => [1,1,1] => 0 = 1 - 1
100 => 101 => [1,1,1] => 0 = 1 - 1
101 => 110 => [2,1] => 1 = 2 - 1
110 => 111 => [3] => 0 = 1 - 1
111 => 111 => [3] => 0 = 1 - 1
0000 => 0001 => [3,1] => 1 = 2 - 1
0001 => 0010 => [2,1,1] => 1 = 2 - 1
0010 => 0101 => [1,1,1,1] => 0 = 1 - 1
0011 => 0101 => [1,1,1,1] => 0 = 1 - 1
0100 => 1001 => [1,2,1] => 2 = 3 - 1
0101 => 1010 => [1,1,1,1] => 0 = 1 - 1
0110 => 1011 => [1,1,2] => 0 = 1 - 1
0111 => 1011 => [1,1,2] => 0 = 1 - 1
1000 => 1001 => [1,2,1] => 2 = 3 - 1
1001 => 1010 => [1,1,1,1] => 0 = 1 - 1
1010 => 1101 => [2,1,1] => 1 = 2 - 1
1011 => 1101 => [2,1,1] => 1 = 2 - 1
1100 => 1101 => [2,1,1] => 1 = 2 - 1
1101 => 1110 => [3,1] => 1 = 2 - 1
1110 => 1111 => [4] => 0 = 1 - 1
1111 => 1111 => [4] => 0 = 1 - 1
Description
The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].
Mp00234: Binary words valleys-to-peaksBinary words
Mp00268: Binary words zeros to flag zerosBinary words
St000875: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0 = 1 - 1
1 => 1 => 1 => 0 = 1 - 1
00 => 01 => 00 => 0 = 1 - 1
01 => 10 => 01 => 0 = 1 - 1
10 => 11 => 11 => 0 = 1 - 1
11 => 11 => 11 => 0 = 1 - 1
000 => 001 => 110 => 1 = 2 - 1
001 => 010 => 100 => 1 = 2 - 1
010 => 101 => 001 => 0 = 1 - 1
011 => 101 => 001 => 0 = 1 - 1
100 => 101 => 001 => 0 = 1 - 1
101 => 110 => 011 => 0 = 1 - 1
110 => 111 => 111 => 0 = 1 - 1
111 => 111 => 111 => 0 = 1 - 1
0000 => 0001 => 0010 => 1 = 2 - 1
0001 => 0010 => 0110 => 1 = 2 - 1
0010 => 0101 => 1100 => 2 = 3 - 1
0011 => 0101 => 1100 => 2 = 3 - 1
0100 => 1001 => 1101 => 1 = 2 - 1
0101 => 1010 => 1001 => 1 = 2 - 1
0110 => 1011 => 0001 => 0 = 1 - 1
0111 => 1011 => 0001 => 0 = 1 - 1
1000 => 1001 => 1101 => 1 = 2 - 1
1001 => 1010 => 1001 => 1 = 2 - 1
1010 => 1101 => 0011 => 0 = 1 - 1
1011 => 1101 => 0011 => 0 = 1 - 1
1100 => 1101 => 0011 => 0 = 1 - 1
1101 => 1110 => 0111 => 0 = 1 - 1
1110 => 1111 => 1111 => 0 = 1 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
Description
The semilength of the longest Dyck word in the Catalan factorisation 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 semilength of the longest Dyck word in this factorisation.
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001323: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> 0 = 1 - 1
1 => [1] => ([],1)
=> 0 = 1 - 1
00 => [2] => ([],2)
=> 0 = 1 - 1
01 => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
10 => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
11 => [2] => ([],2)
=> 0 = 1 - 1
000 => [3] => ([],3)
=> 0 = 1 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
011 => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
100 => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
111 => [3] => ([],3)
=> 0 = 1 - 1
0000 => [4] => ([],4)
=> 0 = 1 - 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0111 => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
1000 => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
1111 => [4] => ([],4)
=> 0 = 1 - 1
Description
The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is $0$ for well covered graphs
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001331: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> 0 = 1 - 1
1 => [1] => ([],1)
=> 0 = 1 - 1
00 => [2] => ([],2)
=> 0 = 1 - 1
01 => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
10 => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
11 => [2] => ([],2)
=> 0 = 1 - 1
000 => [3] => ([],3)
=> 0 = 1 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
011 => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
100 => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
111 => [3] => ([],3)
=> 0 = 1 - 1
0000 => [4] => ([],4)
=> 0 = 1 - 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0111 => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
1000 => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
1111 => [4] => ([],4)
=> 0 = 1 - 1
Description
The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001336: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> 0 = 1 - 1
1 => [1] => ([],1)
=> 0 = 1 - 1
00 => [2] => ([],2)
=> 0 = 1 - 1
01 => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
10 => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
11 => [2] => ([],2)
=> 0 = 1 - 1
000 => [3] => ([],3)
=> 0 = 1 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
011 => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
100 => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
111 => [3] => ([],3)
=> 0 = 1 - 1
0000 => [4] => ([],4)
=> 0 = 1 - 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0111 => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
1000 => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
1111 => [4] => ([],4)
=> 0 = 1 - 1
Description
The minimal number of vertices in a graph whose complement is triangle-free.
The following 215 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001689The number of celebrities in a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000346The number of coarsenings of a partition. St000388The number of orbits of vertices of a graph under automorphisms. St000482The (zero)-forcing number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000570The Edelman-Greene number of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001238The 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. St001312Number of parabolic noncrossing partitions indexed by the composition. St001352The number of internal nodes in the modular decomposition of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001616The number of neutral elements in a lattice. St001642The Prague dimension of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001814The number of partitions interlacing the given partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000010The length of the partition. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000147The largest part of an integer partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000204The number of internal nodes of a binary tree. St000225Difference between largest and smallest parts in a partition. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000463The number of admissible inversions of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000921The number of internal inversions of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001022Number of simple modules with projective dimension 3 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 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. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001565The number of arithmetic progressions of length 2 in a permutation. St001584The area statistic between a Dyck path and its bounce path. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001622The number of join-irreducible elements of a lattice. St001638The book thickness 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. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000619The number of cyclic descents of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St000369The dinv deficit of a Dyck path. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000931The number of occurrences of the pattern UUU in a Dyck path. 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,. St001960The number of descents of a permutation minus one if its first entry is not one. St001128The exponens consonantiae of a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000993The multiplicity of the largest part of an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001282The number of graphs with the same chromatic polynomial. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. 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. St001656The monophonic position number of a graph. St001871The number of triconnected components of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. 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. St000260The radius of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001118The acyclic chromatic index of a graph. St001271The competition number of a graph. 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. 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. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000939The number of characters of the symmetric group whose value on the partition is positive. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001621The number of atoms of a lattice. St001812The biclique partition number of a graph. St001625The Möbius invariant of a lattice. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. 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. St001722The number of minimal chains with small intervals between a binary word and the top element. St000284The Plancherel distribution on integer partitions. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001060The distinguishing index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.