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Your data matches 76 different statistics following compositions of up to 3 maps.
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Matching statistic: St000628
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 1 = 2 - 1
111 => 0 = 1 - 1
0000 => 0 = 1 - 1
0001 => 1 = 2 - 1
0010 => 1 = 2 - 1
0011 => 2 = 3 - 1
0100 => 1 = 2 - 1
0101 => 1 = 2 - 1
0110 => 1 = 2 - 1
0111 => 1 = 2 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 1 = 2 - 1
1100 => 2 = 3 - 1
1101 => 1 = 2 - 1
1110 => 1 = 2 - 1
1111 => 0 = 1 - 1
00000 => 0 = 1 - 1
00001 => 1 = 2 - 1
00010 => 1 = 2 - 1
00011 => 2 = 3 - 1
00100 => 1 = 2 - 1
00101 => 1 = 2 - 1
00110 => 2 = 3 - 1
00111 => 2 = 3 - 1
01000 => 1 = 2 - 1
01001 => 1 = 2 - 1
01010 => 1 = 2 - 1
01011 => 1 = 2 - 1
01100 => 2 = 3 - 1
01101 => 1 = 2 - 1
01110 => 1 = 2 - 1
01111 => 1 = 2 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 2 = 3 - 1
Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000955
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> 1
1 => [1,1] => [1,0,1,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> 2
11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St001165
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> 1
1 => [1] => [1,0]
=> [1,0]
=> 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001471
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> 1
1 => [1] => [1,0]
=> [1,0]
=> 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
Description
The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0 => 0 = 1 - 1
1 => 1 => 1 => 0 => 0 = 1 - 1
00 => 01 => 10 => 01 => 1 = 2 - 1
01 => 10 => 00 => 11 => 1 = 2 - 1
10 => 11 => 11 => 00 => 0 = 1 - 1
11 => 11 => 11 => 00 => 0 = 1 - 1
000 => 001 => 101 => 010 => 1 = 2 - 1
001 => 010 => 000 => 111 => 1 = 2 - 1
010 => 101 => 100 => 011 => 1 = 2 - 1
011 => 101 => 100 => 011 => 1 = 2 - 1
100 => 101 => 100 => 011 => 1 = 2 - 1
101 => 110 => 001 => 110 => 1 = 2 - 1
110 => 111 => 111 => 000 => 0 = 1 - 1
111 => 111 => 111 => 000 => 0 = 1 - 1
0000 => 0001 => 1011 => 0100 => 1 = 2 - 1
0001 => 0010 => 0001 => 1110 => 1 = 2 - 1
0010 => 0101 => 1000 => 0111 => 1 = 2 - 1
0011 => 0101 => 1000 => 0111 => 1 = 2 - 1
0100 => 1001 => 1010 => 0101 => 2 = 3 - 1
0101 => 1010 => 0000 => 1111 => 1 = 2 - 1
0110 => 1011 => 1100 => 0011 => 1 = 2 - 1
0111 => 1011 => 1100 => 0011 => 1 = 2 - 1
1000 => 1001 => 1010 => 0101 => 2 = 3 - 1
1001 => 1010 => 0000 => 1111 => 1 = 2 - 1
1010 => 1101 => 1001 => 0110 => 1 = 2 - 1
1011 => 1101 => 1001 => 0110 => 1 = 2 - 1
1100 => 1101 => 1001 => 0110 => 1 = 2 - 1
1101 => 1110 => 0011 => 1100 => 1 = 2 - 1
1110 => 1111 => 1111 => 0000 => 0 = 1 - 1
1111 => 1111 => 1111 => 0000 => 0 = 1 - 1
00000 => 00001 => 10111 => 01000 => 1 = 2 - 1
00001 => 00010 => 00011 => 11100 => 1 = 2 - 1
00010 => 00101 => 10001 => 01110 => 1 = 2 - 1
00011 => 00101 => 10001 => 01110 => 1 = 2 - 1
00100 => 01001 => 10100 => 01011 => 2 = 3 - 1
00101 => 01010 => 00000 => 11111 => 1 = 2 - 1
00110 => 01011 => 11000 => 00111 => 1 = 2 - 1
00111 => 01011 => 11000 => 00111 => 1 = 2 - 1
01000 => 10001 => 10110 => 01001 => 2 = 3 - 1
01001 => 10010 => 00010 => 11101 => 2 = 3 - 1
01010 => 10101 => 10000 => 01111 => 1 = 2 - 1
01011 => 10101 => 10000 => 01111 => 1 = 2 - 1
01100 => 10101 => 10000 => 01111 => 1 = 2 - 1
01101 => 10110 => 00100 => 11011 => 2 = 3 - 1
01110 => 10111 => 11100 => 00011 => 1 = 2 - 1
01111 => 10111 => 11100 => 00011 => 1 = 2 - 1
10000 => 10001 => 10110 => 01001 => 2 = 3 - 1
10001 => 10010 => 00010 => 11101 => 2 = 3 - 1
10010 => 10101 => 10000 => 01111 => 1 = 2 - 1
10011 => 10101 => 10000 => 01111 => 1 = 2 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000985
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => ([],1)
=> 0 = 1 - 1
1 => 1 => [1] => ([],1)
=> 0 = 1 - 1
00 => 01 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
01 => 10 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
10 => 11 => [2] => ([],2)
=> 0 = 1 - 1
11 => 11 => [2] => ([],2)
=> 0 = 1 - 1
000 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
010 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
011 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
100 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
101 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
110 => 111 => [3] => ([],3)
=> 0 = 1 - 1
111 => 111 => [3] => ([],3)
=> 0 = 1 - 1
0000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
0001 => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0010 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0011 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0100 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0110 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0111 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1000 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
1001 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1010 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1011 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1100 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1101 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
1110 => 1111 => [4] => ([],4)
=> 0 = 1 - 1
1111 => 1111 => [4] => ([],4)
=> 0 = 1 - 1
00000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
00001 => 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
00010 => 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 = 2 - 1
00011 => 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 = 2 - 1
00100 => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
00101 => 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 = 2 - 1
00110 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
00111 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
01000 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
01001 => 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
01010 => 10101 => [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 = 2 - 1
01011 => 10101 => [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 = 2 - 1
01100 => 10101 => [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 = 2 - 1
01101 => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
01110 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
01111 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
10000 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
10001 => 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
10010 => 10101 => [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 = 2 - 1
10011 => 10101 => [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 = 2 - 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001354
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001354: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001354: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => ([],1)
=> 0 = 1 - 1
1 => 1 => [1] => ([],1)
=> 0 = 1 - 1
00 => 01 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
01 => 10 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
10 => 11 => [2] => ([],2)
=> 0 = 1 - 1
11 => 11 => [2] => ([],2)
=> 0 = 1 - 1
000 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
010 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
011 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
100 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
101 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
110 => 111 => [3] => ([],3)
=> 0 = 1 - 1
111 => 111 => [3] => ([],3)
=> 0 = 1 - 1
0000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
0001 => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0010 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0011 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0100 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0110 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
0111 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1000 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
1001 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1010 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1011 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1100 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
1101 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
1110 => 1111 => [4] => ([],4)
=> 0 = 1 - 1
1111 => 1111 => [4] => ([],4)
=> 0 = 1 - 1
00000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
00001 => 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
00010 => 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 = 2 - 1
00011 => 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 = 2 - 1
00100 => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
00101 => 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 = 2 - 1
00110 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
00111 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
01000 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
01001 => 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
01010 => 10101 => [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 = 2 - 1
01011 => 10101 => [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 = 2 - 1
01100 => 10101 => [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 = 2 - 1
01101 => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
01110 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
01111 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
10000 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
10001 => 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
10010 => 10101 => [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 = 2 - 1
10011 => 10101 => [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 = 2 - 1
Description
The number of series nodes in the modular decomposition of a graph.
Matching statistic: St001506
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
1 => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
Description
Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.
Matching statistic: St001198
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? ∊ {1,1}
1 => [1] => [1,0]
=> [1,0]
=> ? ∊ {1,1}
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1}
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1}
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1}
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1}
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1}
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1}
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1}
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1}
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
Description
The 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 $eA$.
Matching statistic: St001206
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? ∊ {1,1}
1 => [1] => [1,0]
=> [1,0]
=> ? ∊ {1,1}
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1}
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1}
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1}
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1}
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1}
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1}
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1}
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1}
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
Description
The 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 $eA$.
The following 66 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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 $eA$. St000260The radius of a connected graph. St001489The maximum of the number of descents and the number of inverse descents. St000455The second largest eigenvalue of a graph if it is integral. St000093The cardinality of a maximal independent set of vertices of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001335The cardinality of a minimal cycle-isolating set of a graph. St000098The chromatic number of a graph. St000097The order of the largest clique of the graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. 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. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001330The hat guessing number of a graph. St001358The largest degree of a regular subgraph of a graph. 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. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. 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. St001642The Prague dimension of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001746The coalition number 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. St000778The metric dimension of a 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. St001060The distinguishing index of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000258The burning number of a graph. St001638The book thickness of a graph. St001674The number of vertices of the largest induced star graph in the 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. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000264The girth of a graph, which is not a tree. 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 $K[x]/(x^n)$.
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