Your data matches 151 different statistics following compositions of up to 3 maps.
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Matching statistic: St000626
(load all 7 compositions to match this statistic)
Mapping
St000626: 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 => 2
10 => 2
11 => 1
000 => 1
001 => 3
010 => 2
011 => 3
100 => 3
101 => 2
110 => 3
111 => 1
0000 => 1
1111 => 1
00000 => 1
11111 => 1
000000 => 1
111111 => 1
Description
The minimal period of a binary word. This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.
Matching statistic: St001415
(load all 5 compositions to match this statistic)
Mapping
Mp00200: Binary words —twist⟶ Binary words
St001415: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1
1 => 0 => 1
00 => 10 => 1
01 => 11 => 2
10 => 00 => 2
11 => 01 => 1
000 => 100 => 1
001 => 101 => 3
010 => 110 => 2
011 => 111 => 3
100 => 000 => 3
101 => 001 => 2
110 => 010 => 3
111 => 011 => 1
0000 => 1000 => 1
1111 => 0111 => 1
00000 => 10000 => 1
11111 => 01111 => 1
000000 => 100000 => 1
111111 => 011111 => 1
Description
The length of the longest palindromic prefix of a binary word.
Matching statistic: St000172
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000388
(load all 2 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St000452
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000452: 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)
 => 2
10 => [1,1] => ([(0,1)],2)
 => 2
11 => [2] => ([],2)
 => 1
000 => [3] => ([],3)
 => 1
001 => [2,1] => ([(0,2),(1,2)],3)
 => 3
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
011 => [1,2] => ([(1,2)],3)
 => 3
100 => [1,2] => ([(1,2)],3)
 => 3
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 3
111 => [3] => ([],3)
 => 1
0000 => [4] => ([],4)
 => 1
1111 => [4] => ([],4)
 => 1
00000 => [5] => ([],5)
 => 1
11111 => [5] => ([],5)
 => 1
000000 => [6] => ([],6)
 => 1
111111 => [6] => ([],6)
 => 1
Description
The number of distinct eigenvalues of a graph.
Matching statistic: St001108
(load all 2 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.
Matching statistic: St001110
(load all 2 compositions to match this statistic)
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001110: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The 3-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-dynamic chromatic number of a graph.
Matching statistic: St001116
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
Matching statistic: St001367
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001367: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The smallest number which does not occur as degree of a vertex in a graph.
Matching statistic: St001581
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => ([],2)
 => 1
1 => ([(0,1)],2)
 => ([],2)
 => 1
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
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)
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
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)
 => 3
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => 1
Description
The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000271The chromatic index of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000501The size of the first part in the decomposition of a permutation. St000722The number of different neighbourhoods in a graph. St000935The number of ordered refinements of an integer partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001389The number of partitions of the same length below the given integer partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001512The minimum rank of a graph. St001725The harmonious chromatic number of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000209Maximum difference of elements in cycles. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001093The detour number of a graph. St001120The length of a longest path in a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001375The pancake length of a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001869The maximum cut size of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000420The number of Dyck paths that are weakly above a Dyck path. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000933The number of multipartitions of sizes given by an integer partition. St000988The orbit size of a permutation under Foata's bijection. St000216The absolute length of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000946The sum of the skew hook positions in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001480The number of simple summands of the module J^2/J^3. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001684The reduced word complexity of a permutation. St000467The hyper-Wiener index of a connected graph. St000682The Grundy value of Welter's game on a binary word. St000455The second largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St001330The hat guessing number of a graph. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a 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 $K[x]/(x^n)$. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000993The multiplicity of the largest part of an integer partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001808The box weight or horizontal decoration of a Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000264The girth of a graph, which is not a tree. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001204Call 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. St001281The normalized isoperimetric number of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001592The maximal number of simple paths between any two different vertices of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000460The hook length of the last cell along the main diagonal of an integer partition. 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. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs 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. St000806The semiperimeter of the associated bargraph. St001060The distinguishing index of a graph. St000464The Schultz index of a connected graph. 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 $eA$. 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 $eA$. 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 $eA$. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St000699The toughness times the least common multiple of 1,. 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.