Your data matches 127 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001489
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 1
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 1
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 1
=> [1] => [1,0]
 => [2,1] => 1
Description
The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000628
(load all 28 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 0 = 1 - 1
00 => 00 => 0 = 1 - 1
01 => 10 => 1 = 2 - 1
10 => 10 => 1 = 2 - 1
11 => 11 => 0 = 1 - 1
000 => 000 => 0 = 1 - 1
001 => 100 => 1 = 2 - 1
010 => 100 => 1 = 2 - 1
011 => 110 => 1 = 2 - 1
100 => 100 => 1 = 2 - 1
101 => 110 => 1 = 2 - 1
110 => 110 => 1 = 2 - 1
111 => 111 => 0 = 1 - 1
0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 1 = 2 - 1
0010 => 1000 => 1 = 2 - 1
0011 => 1010 => 1 = 2 - 1
0100 => 1000 => 1 = 2 - 1
0101 => 1100 => 2 = 3 - 1
0110 => 1010 => 1 = 2 - 1
0111 => 1110 => 1 = 2 - 1
1000 => 1000 => 1 = 2 - 1
1001 => 1010 => 1 = 2 - 1
1010 => 1100 => 2 = 3 - 1
1011 => 1110 => 1 = 2 - 1
1100 => 1010 => 1 = 2 - 1
1101 => 1110 => 1 = 2 - 1
1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 0 = 1 - 1
=> ? => ? = 1 - 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: St000291
(load all 2 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 1 - 1
1 => 1 => 1 => 0 = 1 - 1
00 => 00 => 00 => 0 = 1 - 1
01 => 10 => 10 => 1 = 2 - 1
10 => 10 => 10 => 1 = 2 - 1
11 => 11 => 11 => 0 = 1 - 1
000 => 000 => 000 => 0 = 1 - 1
001 => 100 => 010 => 1 = 2 - 1
010 => 100 => 010 => 1 = 2 - 1
011 => 110 => 110 => 1 = 2 - 1
100 => 100 => 010 => 1 = 2 - 1
101 => 110 => 110 => 1 = 2 - 1
110 => 110 => 110 => 1 = 2 - 1
111 => 111 => 111 => 0 = 1 - 1
0000 => 0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 0010 => 1 = 2 - 1
0010 => 1000 => 0010 => 1 = 2 - 1
0011 => 1010 => 0110 => 1 = 2 - 1
0100 => 1000 => 0010 => 1 = 2 - 1
0101 => 1100 => 1010 => 2 = 3 - 1
0110 => 1010 => 0110 => 1 = 2 - 1
0111 => 1110 => 1110 => 1 = 2 - 1
1000 => 1000 => 0010 => 1 = 2 - 1
1001 => 1010 => 0110 => 1 = 2 - 1
1010 => 1100 => 1010 => 2 = 3 - 1
1011 => 1110 => 1110 => 1 = 2 - 1
1100 => 1010 => 0110 => 1 = 2 - 1
1101 => 1110 => 1110 => 1 = 2 - 1
1110 => 1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
=> ? => ? => ? = 1 - 1
Description
The number of descents of a binary word.
Matching statistic: St000875
(load all 2 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 1 - 1
1 => 1 => 1 => 0 = 1 - 1
00 => 00 => 00 => 0 = 1 - 1
01 => 10 => 10 => 1 = 2 - 1
10 => 10 => 10 => 1 = 2 - 1
11 => 11 => 11 => 0 = 1 - 1
000 => 000 => 000 => 0 = 1 - 1
001 => 100 => 010 => 1 = 2 - 1
010 => 100 => 010 => 1 = 2 - 1
011 => 110 => 110 => 1 = 2 - 1
100 => 100 => 010 => 1 = 2 - 1
101 => 110 => 110 => 1 = 2 - 1
110 => 110 => 110 => 1 = 2 - 1
111 => 111 => 111 => 0 = 1 - 1
0000 => 0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 0010 => 1 = 2 - 1
0010 => 1000 => 0010 => 1 = 2 - 1
0011 => 1010 => 0110 => 1 = 2 - 1
0100 => 1000 => 0010 => 1 = 2 - 1
0101 => 1100 => 1010 => 2 = 3 - 1
0110 => 1010 => 0110 => 1 = 2 - 1
0111 => 1110 => 1110 => 1 = 2 - 1
1000 => 1000 => 0010 => 1 = 2 - 1
1001 => 1010 => 0110 => 1 = 2 - 1
1010 => 1100 => 1010 => 2 = 3 - 1
1011 => 1110 => 1110 => 1 = 2 - 1
1100 => 1010 => 0110 => 1 = 2 - 1
1101 => 1110 => 1110 => 1 = 2 - 1
1110 => 1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
=> ? => ? => ? = 1 - 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001420
(load all 7 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001420: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 1 - 1
1 => 1 => 1 => 0 = 1 - 1
00 => 00 => 00 => 0 = 1 - 1
01 => 10 => 10 => 1 = 2 - 1
10 => 10 => 10 => 1 = 2 - 1
11 => 11 => 11 => 0 = 1 - 1
000 => 000 => 000 => 0 = 1 - 1
001 => 100 => 010 => 1 = 2 - 1
010 => 100 => 010 => 1 = 2 - 1
011 => 110 => 110 => 1 = 2 - 1
100 => 100 => 010 => 1 = 2 - 1
101 => 110 => 110 => 1 = 2 - 1
110 => 110 => 110 => 1 = 2 - 1
111 => 111 => 111 => 0 = 1 - 1
0000 => 0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 0010 => 1 = 2 - 1
0010 => 1000 => 0010 => 1 = 2 - 1
0011 => 1010 => 0110 => 1 = 2 - 1
0100 => 1000 => 0010 => 1 = 2 - 1
0101 => 1100 => 1010 => 2 = 3 - 1
0110 => 1010 => 0110 => 1 = 2 - 1
0111 => 1110 => 1110 => 1 = 2 - 1
1000 => 1000 => 0010 => 1 = 2 - 1
1001 => 1010 => 0110 => 1 = 2 - 1
1010 => 1100 => 1010 => 2 = 3 - 1
1011 => 1110 => 1110 => 1 = 2 - 1
1100 => 1010 => 0110 => 1 = 2 - 1
1101 => 1110 => 1110 => 1 = 2 - 1
1110 => 1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
=> ? => ? => ? = 1 - 1
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Matching statistic: St001421
(load all 2 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001421: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 1 - 1
1 => 1 => 1 => 0 = 1 - 1
00 => 00 => 00 => 0 = 1 - 1
01 => 10 => 10 => 1 = 2 - 1
10 => 10 => 10 => 1 = 2 - 1
11 => 11 => 11 => 0 = 1 - 1
000 => 000 => 000 => 0 = 1 - 1
001 => 100 => 010 => 1 = 2 - 1
010 => 100 => 010 => 1 = 2 - 1
011 => 110 => 110 => 1 = 2 - 1
100 => 100 => 010 => 1 = 2 - 1
101 => 110 => 110 => 1 = 2 - 1
110 => 110 => 110 => 1 = 2 - 1
111 => 111 => 111 => 0 = 1 - 1
0000 => 0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 0010 => 1 = 2 - 1
0010 => 1000 => 0010 => 1 = 2 - 1
0011 => 1010 => 0110 => 1 = 2 - 1
0100 => 1000 => 0010 => 1 = 2 - 1
0101 => 1100 => 1010 => 2 = 3 - 1
0110 => 1010 => 0110 => 1 = 2 - 1
0111 => 1110 => 1110 => 1 = 2 - 1
1000 => 1000 => 0010 => 1 = 2 - 1
1001 => 1010 => 0110 => 1 = 2 - 1
1010 => 1100 => 1010 => 2 = 3 - 1
1011 => 1110 => 1110 => 1 = 2 - 1
1100 => 1010 => 0110 => 1 = 2 - 1
1101 => 1110 => 1110 => 1 = 2 - 1
1110 => 1110 => 1110 => 1 = 2 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
=> ? => ? => ? = 1 - 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001165
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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]
 => [1,1,0,0]
 => 1
01 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
10 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
11 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1
000 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
001 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
110 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
111 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
=> [] => ?
 => ?
 => ? = 1
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)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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]
 => [1,1,0,0]
 => 1
01 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
10 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
11 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1
000 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
001 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
110 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
111 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
=> [] => ?
 => ?
 => ? = 1
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: St000292
(load all 3 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 => 0 = 1 - 1
1 => 1 => 1 => 1 => 0 = 1 - 1
00 => 00 => 00 => 00 => 0 = 1 - 1
01 => 10 => 10 => 01 => 1 = 2 - 1
10 => 10 => 10 => 01 => 1 = 2 - 1
11 => 11 => 11 => 11 => 0 = 1 - 1
000 => 000 => 000 => 000 => 0 = 1 - 1
001 => 100 => 010 => 001 => 1 = 2 - 1
010 => 100 => 010 => 001 => 1 = 2 - 1
011 => 110 => 110 => 011 => 1 = 2 - 1
100 => 100 => 010 => 001 => 1 = 2 - 1
101 => 110 => 110 => 011 => 1 = 2 - 1
110 => 110 => 110 => 011 => 1 = 2 - 1
111 => 111 => 111 => 111 => 0 = 1 - 1
0000 => 0000 => 0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 0010 => 0001 => 1 = 2 - 1
0010 => 1000 => 0010 => 0001 => 1 = 2 - 1
0011 => 1010 => 0110 => 0011 => 1 = 2 - 1
0100 => 1000 => 0010 => 0001 => 1 = 2 - 1
0101 => 1100 => 1010 => 0101 => 2 = 3 - 1
0110 => 1010 => 0110 => 0011 => 1 = 2 - 1
0111 => 1110 => 1110 => 0111 => 1 = 2 - 1
1000 => 1000 => 0010 => 0001 => 1 = 2 - 1
1001 => 1010 => 0110 => 0011 => 1 = 2 - 1
1010 => 1100 => 1010 => 0101 => 2 = 3 - 1
1011 => 1110 => 1110 => 0111 => 1 = 2 - 1
1100 => 1010 => 0110 => 0011 => 1 = 2 - 1
1101 => 1110 => 1110 => 0111 => 1 = 2 - 1
1110 => 1110 => 1110 => 0111 => 1 = 2 - 1
1111 => 1111 => 1111 => 1111 => 0 = 1 - 1
=> ? => ? => ? => ? = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000387
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000387: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
1 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
00 => [2] => ([],2)
 => ([],1)
 => 0 = 1 - 1
01 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
10 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
11 => [2] => ([],2)
 => ([],1)
 => 0 = 1 - 1
000 => [3] => ([],3)
 => ([],1)
 => 0 = 1 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
011 => [1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
100 => [1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
111 => [3] => ([],3)
 => ([],1)
 => 0 = 1 - 1
0000 => [4] => ([],4)
 => ([],1)
 => 0 = 1 - 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
0111 => [1,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
1000 => [1,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
1111 => [4] => ([],4)
 => ([],1)
 => 0 = 1 - 1
=> [] => ?
 => ?
 => ? = 1 - 1
Description
The matching number of a graph. For a graph $G$, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in $G$.
The following 117 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000691The number of changes 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. St001071The beta invariant of the graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001665The number of pure excedances of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000098The chromatic number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected 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. 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. St000259The diameter of a connected 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. 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. St001323The independence gap of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001569The maximal modular displacement of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001624The breadth of a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001651The Frankl number of a lattice. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001118The acyclic chromatic index of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition.