Your data matches 145 different statistics following compositions of up to 3 maps.
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Matching statistic: St000630
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 1 = 2 - 1
1 => 1 => 1 => 1 = 2 - 1
00 => 00 => 00 => 1 = 2 - 1
01 => 10 => 01 => 2 = 3 - 1
10 => 10 => 01 => 2 = 3 - 1
11 => 11 => 11 => 1 = 2 - 1
000 => 000 => 000 => 1 = 2 - 1
001 => 100 => 001 => 2 = 3 - 1
010 => 100 => 001 => 2 = 3 - 1
011 => 110 => 011 => 2 = 3 - 1
100 => 100 => 001 => 2 = 3 - 1
101 => 110 => 011 => 2 = 3 - 1
110 => 110 => 011 => 2 = 3 - 1
111 => 111 => 111 => 1 = 2 - 1
0000 => 0000 => 0000 => 1 = 2 - 1
0001 => 1000 => 0001 => 2 = 3 - 1
0010 => 1000 => 0001 => 2 = 3 - 1
0011 => 1010 => 0011 => 2 = 3 - 1
0100 => 1000 => 0001 => 2 = 3 - 1
0101 => 1100 => 0011 => 2 = 3 - 1
0110 => 1010 => 0011 => 2 = 3 - 1
0111 => 1110 => 0111 => 2 = 3 - 1
1000 => 1000 => 0001 => 2 = 3 - 1
1001 => 1010 => 0011 => 2 = 3 - 1
1010 => 1100 => 0011 => 2 = 3 - 1
1011 => 1110 => 0111 => 2 = 3 - 1
1100 => 1010 => 0011 => 2 = 3 - 1
1101 => 1110 => 0111 => 2 = 3 - 1
1110 => 1110 => 0111 => 2 = 3 - 1
1111 => 1111 => 1111 => 1 = 2 - 1
00000 => 00000 => 00000 => 1 = 2 - 1
00001 => 10000 => 00001 => 2 = 3 - 1
00010 => 10000 => 00001 => 2 = 3 - 1
00011 => 10010 => 00011 => 2 = 3 - 1
00100 => 10000 => 00001 => 2 = 3 - 1
00101 => 11000 => 00011 => 2 = 3 - 1
00110 => 10010 => 00011 => 2 = 3 - 1
00111 => 10110 => 00111 => 2 = 3 - 1
01000 => 10000 => 00001 => 2 = 3 - 1
01001 => 11000 => 00011 => 2 = 3 - 1
01010 => 11000 => 00011 => 2 = 3 - 1
01011 => 11100 => 00111 => 2 = 3 - 1
01100 => 10010 => 00011 => 2 = 3 - 1
01101 => 11100 => 00111 => 2 = 3 - 1
01110 => 10110 => 00111 => 2 = 3 - 1
01111 => 11110 => 01111 => 2 = 3 - 1
10000 => 10000 => 00001 => 2 = 3 - 1
10001 => 10010 => 00011 => 2 = 3 - 1
10010 => 11000 => 00011 => 2 = 3 - 1
10011 => 10110 => 00111 => 2 = 3 - 1
Description
The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St000983
(load all 2 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 1 = 2 - 1
1 => 1 => 1 => 1 = 2 - 1
00 => 00 => 00 => 1 = 2 - 1
01 => 10 => 01 => 2 = 3 - 1
10 => 10 => 01 => 2 = 3 - 1
11 => 11 => 11 => 1 = 2 - 1
000 => 000 => 000 => 1 = 2 - 1
001 => 100 => 001 => 2 = 3 - 1
010 => 100 => 001 => 2 = 3 - 1
011 => 110 => 011 => 2 = 3 - 1
100 => 100 => 001 => 2 = 3 - 1
101 => 110 => 011 => 2 = 3 - 1
110 => 110 => 011 => 2 = 3 - 1
111 => 111 => 111 => 1 = 2 - 1
0000 => 0000 => 0000 => 1 = 2 - 1
0001 => 1000 => 0001 => 2 = 3 - 1
0010 => 1000 => 0001 => 2 = 3 - 1
0011 => 1010 => 0011 => 2 = 3 - 1
0100 => 1000 => 0001 => 2 = 3 - 1
0101 => 1100 => 0011 => 2 = 3 - 1
0110 => 1010 => 0011 => 2 = 3 - 1
0111 => 1110 => 0111 => 2 = 3 - 1
1000 => 1000 => 0001 => 2 = 3 - 1
1001 => 1010 => 0011 => 2 = 3 - 1
1010 => 1100 => 0011 => 2 = 3 - 1
1011 => 1110 => 0111 => 2 = 3 - 1
1100 => 1010 => 0011 => 2 = 3 - 1
1101 => 1110 => 0111 => 2 = 3 - 1
1110 => 1110 => 0111 => 2 = 3 - 1
1111 => 1111 => 1111 => 1 = 2 - 1
00000 => 00000 => 00000 => 1 = 2 - 1
00001 => 10000 => 00001 => 2 = 3 - 1
00010 => 10000 => 00001 => 2 = 3 - 1
00011 => 10010 => 00011 => 2 = 3 - 1
00100 => 10000 => 00001 => 2 = 3 - 1
00101 => 11000 => 00011 => 2 = 3 - 1
00110 => 10010 => 00011 => 2 = 3 - 1
00111 => 10110 => 00111 => 2 = 3 - 1
01000 => 10000 => 00001 => 2 = 3 - 1
01001 => 11000 => 00011 => 2 = 3 - 1
01010 => 11000 => 00011 => 2 = 3 - 1
01011 => 11100 => 00111 => 2 = 3 - 1
01100 => 10010 => 00011 => 2 = 3 - 1
01101 => 11100 => 00111 => 2 = 3 - 1
01110 => 10110 => 00111 => 2 = 3 - 1
01111 => 11110 => 01111 => 2 = 3 - 1
10000 => 10000 => 00001 => 2 = 3 - 1
10001 => 10010 => 00011 => 2 = 3 - 1
10010 => 11000 => 00011 => 2 = 3 - 1
10011 => 10110 => 00111 => 2 = 3 - 1
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St001239
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: 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]
 => 2 = 3 - 1
10 => [1,1] => [1,0,1,0]
 => 2 = 3 - 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]
 => 2 = 3 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 2 = 3 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 2 = 3 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 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
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 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
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 3 - 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001261
(load all 6 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => 1 = 2 - 1
1 => [1] => ([],1)
 => 1 = 2 - 1
00 => [2] => ([],2)
 => 1 = 2 - 1
01 => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
10 => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
11 => [2] => ([],2)
 => 1 = 2 - 1
000 => [3] => ([],3)
 => 1 = 2 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
011 => [1,2] => ([(1,2)],3)
 => 2 = 3 - 1
100 => [1,2] => ([(1,2)],3)
 => 2 = 3 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
111 => [3] => ([],3)
 => 1 = 2 - 1
0000 => [4] => ([],4)
 => 1 = 2 - 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)
 => 2 = 3 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 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)
 => 2 = 3 - 1
0111 => [1,3] => ([(2,3)],4)
 => 2 = 3 - 1
1000 => [1,3] => ([(2,3)],4)
 => 2 = 3 - 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 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)
 => 2 = 3 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
1111 => [4] => ([],4)
 => 1 = 2 - 1
00000 => [5] => ([],5)
 => 1 = 2 - 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
00111 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
01111 => [1,4] => ([(3,4)],5)
 => 2 = 3 - 1
10000 => [1,4] => ([(3,4)],5)
 => 2 = 3 - 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000292
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 2 - 2
1 => 1 => 1 => 0 = 2 - 2
00 => 00 => 00 => 0 = 2 - 2
01 => 10 => 01 => 1 = 3 - 2
10 => 10 => 01 => 1 = 3 - 2
11 => 11 => 11 => 0 = 2 - 2
000 => 000 => 000 => 0 = 2 - 2
001 => 100 => 001 => 1 = 3 - 2
010 => 100 => 001 => 1 = 3 - 2
011 => 110 => 011 => 1 = 3 - 2
100 => 100 => 001 => 1 = 3 - 2
101 => 110 => 011 => 1 = 3 - 2
110 => 110 => 011 => 1 = 3 - 2
111 => 111 => 111 => 0 = 2 - 2
0000 => 0000 => 0000 => 0 = 2 - 2
0001 => 1000 => 0001 => 1 = 3 - 2
0010 => 1000 => 0001 => 1 = 3 - 2
0011 => 1010 => 0011 => 1 = 3 - 2
0100 => 1000 => 0001 => 1 = 3 - 2
0101 => 1100 => 0011 => 1 = 3 - 2
0110 => 1010 => 0011 => 1 = 3 - 2
0111 => 1110 => 0111 => 1 = 3 - 2
1000 => 1000 => 0001 => 1 = 3 - 2
1001 => 1010 => 0011 => 1 = 3 - 2
1010 => 1100 => 0011 => 1 = 3 - 2
1011 => 1110 => 0111 => 1 = 3 - 2
1100 => 1010 => 0011 => 1 = 3 - 2
1101 => 1110 => 0111 => 1 = 3 - 2
1110 => 1110 => 0111 => 1 = 3 - 2
1111 => 1111 => 1111 => 0 = 2 - 2
00000 => 00000 => 00000 => 0 = 2 - 2
00001 => 10000 => 00001 => 1 = 3 - 2
00010 => 10000 => 00001 => 1 = 3 - 2
00011 => 10010 => 00011 => 1 = 3 - 2
00100 => 10000 => 00001 => 1 = 3 - 2
00101 => 11000 => 00011 => 1 = 3 - 2
00110 => 10010 => 00011 => 1 = 3 - 2
00111 => 10110 => 00111 => 1 = 3 - 2
01000 => 10000 => 00001 => 1 = 3 - 2
01001 => 11000 => 00011 => 1 = 3 - 2
01010 => 11000 => 00011 => 1 = 3 - 2
01011 => 11100 => 00111 => 1 = 3 - 2
01100 => 10010 => 00011 => 1 = 3 - 2
01101 => 11100 => 00111 => 1 = 3 - 2
01110 => 10110 => 00111 => 1 = 3 - 2
01111 => 11110 => 01111 => 1 = 3 - 2
10000 => 10000 => 00001 => 1 = 3 - 2
10001 => 10010 => 00011 => 1 = 3 - 2
10010 => 11000 => 00011 => 1 = 3 - 2
10011 => 10110 => 00111 => 1 = 3 - 2
Description
The number of ascents of a binary word.
Matching statistic: St000535
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => 0 = 2 - 2
1 => [1] => ([],1)
 => 0 = 2 - 2
00 => [2] => ([],2)
 => 0 = 2 - 2
01 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
10 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
11 => [2] => ([],2)
 => 0 = 2 - 2
000 => [3] => ([],3)
 => 0 = 2 - 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
011 => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
100 => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
111 => [3] => ([],3)
 => 0 = 2 - 2
0000 => [4] => ([],4)
 => 0 = 2 - 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0111 => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
1000 => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
1111 => [4] => ([],4)
 => 0 = 2 - 2
00000 => [5] => ([],5)
 => 0 = 2 - 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00111 => [2,3] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01111 => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
10000 => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
Description
The rank-width of a graph.
Matching statistic: St000691
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 2 - 2
1 => 1 => 1 => 0 = 2 - 2
00 => 00 => 00 => 0 = 2 - 2
01 => 10 => 01 => 1 = 3 - 2
10 => 10 => 01 => 1 = 3 - 2
11 => 11 => 11 => 0 = 2 - 2
000 => 000 => 000 => 0 = 2 - 2
001 => 100 => 001 => 1 = 3 - 2
010 => 100 => 001 => 1 = 3 - 2
011 => 110 => 011 => 1 = 3 - 2
100 => 100 => 001 => 1 = 3 - 2
101 => 110 => 011 => 1 = 3 - 2
110 => 110 => 011 => 1 = 3 - 2
111 => 111 => 111 => 0 = 2 - 2
0000 => 0000 => 0000 => 0 = 2 - 2
0001 => 1000 => 0001 => 1 = 3 - 2
0010 => 1000 => 0001 => 1 = 3 - 2
0011 => 1010 => 0011 => 1 = 3 - 2
0100 => 1000 => 0001 => 1 = 3 - 2
0101 => 1100 => 0011 => 1 = 3 - 2
0110 => 1010 => 0011 => 1 = 3 - 2
0111 => 1110 => 0111 => 1 = 3 - 2
1000 => 1000 => 0001 => 1 = 3 - 2
1001 => 1010 => 0011 => 1 = 3 - 2
1010 => 1100 => 0011 => 1 = 3 - 2
1011 => 1110 => 0111 => 1 = 3 - 2
1100 => 1010 => 0011 => 1 = 3 - 2
1101 => 1110 => 0111 => 1 = 3 - 2
1110 => 1110 => 0111 => 1 = 3 - 2
1111 => 1111 => 1111 => 0 = 2 - 2
00000 => 00000 => 00000 => 0 = 2 - 2
00001 => 10000 => 00001 => 1 = 3 - 2
00010 => 10000 => 00001 => 1 = 3 - 2
00011 => 10010 => 00011 => 1 = 3 - 2
00100 => 10000 => 00001 => 1 = 3 - 2
00101 => 11000 => 00011 => 1 = 3 - 2
00110 => 10010 => 00011 => 1 = 3 - 2
00111 => 10110 => 00111 => 1 = 3 - 2
01000 => 10000 => 00001 => 1 = 3 - 2
01001 => 11000 => 00011 => 1 = 3 - 2
01010 => 11000 => 00011 => 1 = 3 - 2
01011 => 11100 => 00111 => 1 = 3 - 2
01100 => 10010 => 00011 => 1 = 3 - 2
01101 => 11100 => 00111 => 1 = 3 - 2
01110 => 10110 => 00111 => 1 = 3 - 2
01111 => 11110 => 01111 => 1 = 3 - 2
10000 => 10000 => 00001 => 1 = 3 - 2
10001 => 10010 => 00011 => 1 = 3 - 2
10010 => 11000 => 00011 => 1 = 3 - 2
10011 => 10110 => 00111 => 1 = 3 - 2
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001192
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 0 = 2 - 2
1 => [1] => [1,0]
 => 0 = 2 - 2
00 => [2] => [1,1,0,0]
 => 0 = 2 - 2
01 => [1,1] => [1,0,1,0]
 => 1 = 3 - 2
10 => [1,1] => [1,0,1,0]
 => 1 = 3 - 2
11 => [2] => [1,1,0,0]
 => 0 = 2 - 2
000 => [3] => [1,1,1,0,0,0]
 => 0 = 2 - 2
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 3 - 2
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 3 - 2
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 3 - 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 3 - 2
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
111 => [3] => [1,1,1,0,0,0]
 => 0 = 2 - 2
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 2 - 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 3 - 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 3 - 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 2 - 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 3 - 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 3 - 2
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 3 - 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 3 - 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 3 - 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 3 - 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 3 - 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 3 - 2
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001333
(load all 6 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => 0 = 2 - 2
1 => [1] => ([],1)
 => 0 = 2 - 2
00 => [2] => ([],2)
 => 0 = 2 - 2
01 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
10 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
11 => [2] => ([],2)
 => 0 = 2 - 2
000 => [3] => ([],3)
 => 0 = 2 - 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
011 => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
100 => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
111 => [3] => ([],3)
 => 0 = 2 - 2
0000 => [4] => ([],4)
 => 0 = 2 - 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0111 => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
1000 => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
1111 => [4] => ([],4)
 => 0 = 2 - 2
00000 => [5] => ([],5)
 => 0 = 2 - 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00111 => [2,3] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01111 => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
10000 => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
Description
The cardinality of a minimal edge-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St001393
(load all 7 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => 0 = 2 - 2
1 => [1] => ([],1)
 => 0 = 2 - 2
00 => [2] => ([],2)
 => 0 = 2 - 2
01 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
10 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
11 => [2] => ([],2)
 => 0 = 2 - 2
000 => [3] => ([],3)
 => 0 = 2 - 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
011 => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
100 => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
111 => [3] => ([],3)
 => 0 = 2 - 2
0000 => [4] => ([],4)
 => 0 = 2 - 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
0111 => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
1000 => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 3 - 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 3 - 2
1111 => [4] => ([],4)
 => 0 = 2 - 2
00000 => [5] => ([],5)
 => 0 = 2 - 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
00111 => [2,3] => ([(2,4),(3,4)],5)
 => 1 = 3 - 2
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
01111 => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
10000 => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 3 - 2
Description
The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001486The number of corners of the ribbon associated with an integer composition. St000013The height of a Dyck path. St000058The order of a permutation. St000258The burning number of a graph. St000259The diameter of a connected graph. St000299The number of nonisomorphic vertex-induced subtrees. St000397The Strahler number of a rooted tree. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000903The number of different parts of an integer composition. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001315The dissociation number of a graph. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000260The radius of a connected graph. St000480The number of lower covers of a partition in dominance order. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000769The major index of a composition regarded as a word. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001512The minimum rank of a graph. St001673The degree of asymmetry of an integer composition. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000442The maximal area to the right of an up step of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000659The number of rises of length at least 2 of a Dyck path. St000730The maximal arc length of a set partition. St000919The number of maximal left branches of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000264The girth of a graph, which is not a tree. St000862The number of parts of the shifted shape of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001060The distinguishing index of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St001092The number of distinct even parts of a partition. St001734The lettericity of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000455The second largest eigenvalue of a graph if it is integral. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000098The chromatic number of a graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001746The coalition number of a graph. St000159The number of distinct parts of the integer partition. St000307The number of rowmotion orbits of a poset. St000544The cop number of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001432The order dimension of the partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001792The arboricity of a graph. St001883The mutual visibility number of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000481The number of upper covers of a partition in dominance order. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001270The bandwidth of a graph. St001323The independence gap of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St000640The rank of the largest boolean interval in a poset. St001642The Prague dimension of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001569The maximal modular displacement of a permutation. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2.