Your data matches 190 different statistics following compositions of up to 3 maps.
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Matching statistic: St000628
(load all 33 compositions to match this statistic)
Mapping
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
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
Mapping
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%
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
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St001716
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001716: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => 1
1 => [1,1] => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
10 => [1,2] => ([(1,2)],3)
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
000 => [4] => ([],4)
 => 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
100 => [1,3] => ([(2,3)],4)
 => 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
0000 => [5] => ([],5)
 => 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1000 => [1,4] => ([(3,4)],5)
 => 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1111 => [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)
 => 3
Description
The 1-improper chromatic number of a graph. This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.
Matching statistic: St000291
(load all 7 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: 100% ●values known / values provided: 100%●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
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 8 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ 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 => 1 => 1 => 0 = 1 - 1
1 => 1 => 1 => 0 = 1 - 1
00 => 01 => 01 => 1 = 2 - 1
01 => 10 => 01 => 1 = 2 - 1
10 => 11 => 11 => 0 = 1 - 1
11 => 11 => 11 => 0 = 1 - 1
000 => 001 => 001 => 1 = 2 - 1
001 => 010 => 001 => 1 = 2 - 1
010 => 101 => 011 => 1 = 2 - 1
011 => 101 => 011 => 1 = 2 - 1
100 => 101 => 011 => 1 = 2 - 1
101 => 110 => 011 => 1 = 2 - 1
110 => 111 => 111 => 0 = 1 - 1
111 => 111 => 111 => 0 = 1 - 1
0000 => 0001 => 0001 => 1 = 2 - 1
0001 => 0010 => 0001 => 1 = 2 - 1
0010 => 0101 => 0101 => 2 = 3 - 1
0011 => 0101 => 0101 => 2 = 3 - 1
0100 => 1001 => 0011 => 1 = 2 - 1
0101 => 1010 => 0011 => 1 = 2 - 1
0110 => 1011 => 0111 => 1 = 2 - 1
0111 => 1011 => 0111 => 1 = 2 - 1
1000 => 1001 => 0011 => 1 = 2 - 1
1001 => 1010 => 0011 => 1 = 2 - 1
1010 => 1101 => 0111 => 1 = 2 - 1
1011 => 1101 => 0111 => 1 = 2 - 1
1100 => 1101 => 0111 => 1 = 2 - 1
1101 => 1110 => 0111 => 1 = 2 - 1
1110 => 1111 => 1111 => 0 = 1 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000875
(load all 3 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: 100% ●values known / values provided: 100%●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
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: St000985
(load all 4 compositions to match this statistic)
Mapping
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)
 => 0 = 1 - 1
1 => [1] => ([],1)
 => 0 = 1 - 1
00 => [2] => ([],2)
 => 0 = 1 - 1
01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
10 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
11 => [2] => ([],2)
 => 0 = 1 - 1
000 => [3] => ([],3)
 => 0 = 1 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
011 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
100 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
111 => [3] => ([],3)
 => 0 = 1 - 1
0000 => [4] => ([],4)
 => 0 = 1 - 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
0111 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
1000 => [1,3] => ([(2,3)],4)
 => 1 = 2 - 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)
 => 1 = 2 - 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
1111 => [4] => ([],4)
 => 0 = 1 - 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001011
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ 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 => [2] => [1,1,0,0]
 => 0 = 1 - 1
01 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
10 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
11 => [2] => [1,1,0,0]
 => 0 = 1 - 1
000 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 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]
 => 1 = 2 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 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]
 => 1 = 2 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001188
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001188: Dyck paths ⟶ 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 => [2] => [1,1,0,0]
 => 0 = 1 - 1
01 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
10 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
11 => [2] => [1,1,0,0]
 => 0 = 1 - 1
000 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 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]
 => 1 = 2 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 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]
 => 1 = 2 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
The 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. Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.
Matching statistic: St001212
(load all 6 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001212: Dyck paths ⟶ 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 => [2] => [1,1,0,0]
 => 0 = 1 - 1
01 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
10 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
11 => [2] => [1,1,0,0]
 => 0 = 1 - 1
000 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
111 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 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]
 => 1 = 2 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 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]
 => 1 = 2 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
The following 180 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001280The number of parts of an integer partition that are at least two. St001354The number of series nodes in the modular decomposition of a graph. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000767The number of runs in an integer composition. St000983The length of the longest alternating subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001792The arboricity of a graph. St001814The number of partitions interlacing the given partition. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000024The number of double up and double down steps of a Dyck path. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000147The largest part of an integer partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000238The number of indices that are not small weak excedances. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000624The normalized sum of the minimal distances to a greater element. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001071The beta invariant of the graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word 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). St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000251The number of nonsingleton blocks of a set partition. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000829The Ulam distance of a permutation to the identity permutation. St000919The number of maximal left branches of a binary tree. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000259The diameter of a connected graph. 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. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000098The chromatic number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected 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. 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. 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. 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. St000741The Colin de Verdière graph invariant. St001569The maximal modular displacement of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. 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. 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 $K[x]/(x^n)$. 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. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001060The distinguishing index of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.