Your data matches 56 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000389
(load all 3 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000389: 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
00 => 00 => 00 => 0
01 => 10 => 01 => 1
10 => 10 => 01 => 1
11 => 11 => 11 => 0
000 => 000 => 000 => 0
001 => 100 => 001 => 1
010 => 100 => 001 => 1
011 => 110 => 011 => 0
100 => 100 => 001 => 1
101 => 110 => 011 => 0
110 => 110 => 011 => 0
111 => 111 => 111 => 1
0000 => 0000 => 0000 => 0
0001 => 1000 => 0001 => 1
0010 => 1000 => 0001 => 1
0011 => 1010 => 0011 => 0
0100 => 1000 => 0001 => 1
0101 => 1100 => 0011 => 0
0110 => 1010 => 0011 => 0
0111 => 1110 => 0111 => 1
1000 => 1000 => 0001 => 1
1001 => 1010 => 0011 => 0
1010 => 1100 => 0011 => 0
1011 => 1110 => 0111 => 1
1100 => 1010 => 0011 => 0
1101 => 1110 => 0111 => 1
1110 => 1110 => 0111 => 1
1111 => 1111 => 1111 => 0
00000 => 00000 => 00000 => 0
00001 => 10000 => 00001 => 1
00010 => 10000 => 00001 => 1
00011 => 10010 => 00011 => 0
00100 => 10000 => 00001 => 1
00101 => 11000 => 00011 => 0
00110 => 10010 => 00011 => 0
00111 => 10110 => 00111 => 1
01000 => 10000 => 00001 => 1
01001 => 11000 => 00011 => 0
01010 => 11000 => 00011 => 0
01011 => 11100 => 00111 => 1
01100 => 10010 => 00011 => 0
01101 => 11100 => 00111 => 1
01110 => 10110 => 00111 => 1
01111 => 11110 => 01111 => 0
10000 => 10000 => 00001 => 1
10001 => 10010 => 00011 => 0
10010 => 11000 => 00011 => 0
10011 => 10110 => 00111 => 1
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St001271
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => 0
1 => [1,1] => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => 0
01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
10 => [1,2] => ([(1,2)],3)
 => 0
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
000 => [4] => ([],4)
 => 0
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => 0
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
100 => [1,3] => ([(2,3)],4)
 => 0
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
0000 => [5] => ([],5)
 => 0
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => 0
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1000 => [1,4] => ([(3,4)],5)
 => 0
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
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)
 => 1
00000 => [6] => ([],6)
 => 0
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => 0
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
10000 => [1,5] => ([(4,5)],6)
 => 0
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The competition number of a graph. The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000326
(load all 4 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 2 = 1 + 1
1 => 1 => 1 => 1 = 0 + 1
00 => 00 => 10 => 1 = 0 + 1
01 => 10 => 01 => 2 = 1 + 1
10 => 10 => 01 => 2 = 1 + 1
11 => 11 => 11 => 1 = 0 + 1
000 => 000 => 010 => 2 = 1 + 1
001 => 100 => 101 => 1 = 0 + 1
010 => 100 => 101 => 1 = 0 + 1
011 => 110 => 011 => 2 = 1 + 1
100 => 100 => 101 => 1 = 0 + 1
101 => 110 => 011 => 2 = 1 + 1
110 => 110 => 011 => 2 = 1 + 1
111 => 111 => 111 => 1 = 0 + 1
0000 => 0000 => 1010 => 1 = 0 + 1
0001 => 1000 => 0101 => 2 = 1 + 1
0010 => 1000 => 0101 => 2 = 1 + 1
0011 => 1010 => 1001 => 1 = 0 + 1
0100 => 1000 => 0101 => 2 = 1 + 1
0101 => 1100 => 1011 => 1 = 0 + 1
0110 => 1010 => 1001 => 1 = 0 + 1
0111 => 1110 => 0111 => 2 = 1 + 1
1000 => 1000 => 0101 => 2 = 1 + 1
1001 => 1010 => 1001 => 1 = 0 + 1
1010 => 1100 => 1011 => 1 = 0 + 1
1011 => 1110 => 0111 => 2 = 1 + 1
1100 => 1010 => 1001 => 1 = 0 + 1
1101 => 1110 => 0111 => 2 = 1 + 1
1110 => 1110 => 0111 => 2 = 1 + 1
1111 => 1111 => 1111 => 1 = 0 + 1
00000 => 00000 => 01010 => 2 = 1 + 1
00001 => 10000 => 10101 => 1 = 0 + 1
00010 => 10000 => 10101 => 1 = 0 + 1
00011 => 10010 => 01101 => 2 = 1 + 1
00100 => 10000 => 10101 => 1 = 0 + 1
00101 => 11000 => 01011 => 2 = 1 + 1
00110 => 10010 => 01101 => 2 = 1 + 1
00111 => 10110 => 10001 => 1 = 0 + 1
01000 => 10000 => 10101 => 1 = 0 + 1
01001 => 11000 => 01011 => 2 = 1 + 1
01010 => 11000 => 01011 => 2 = 1 + 1
01011 => 11100 => 10111 => 1 = 0 + 1
01100 => 10010 => 01101 => 2 = 1 + 1
01101 => 11100 => 10111 => 1 = 0 + 1
01110 => 10110 => 10001 => 1 = 0 + 1
01111 => 11110 => 01111 => 2 = 1 + 1
10000 => 10000 => 10101 => 1 = 0 + 1
10001 => 10010 => 01101 => 2 = 1 + 1
10010 => 11000 => 01011 => 2 = 1 + 1
10011 => 10110 => 10001 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000481
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => [1,1]
 => 1
1 => [1,1] => ([(0,1)],2)
 => [2]
 => 0
00 => [3] => ([],3)
 => [1,1,1]
 => 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 0
10 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
000 => [4] => ([],4)
 => [1,1,1,1]
 => 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 0
010 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
100 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
0000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
0100 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1000 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
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)
 => [5]
 => 0
00000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 0
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
01000 => [2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 1
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
10000 => [1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000932
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
0111 => [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
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
1111 => [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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 0
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 0
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000986
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => ([],1)
 => 1
1 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
00 => [3] => ([],3)
 => ([],1)
 => 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
10 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
000 => [4] => ([],4)
 => ([],1)
 => 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
100 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
111 => [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)
 => 0
0000 => [5] => ([],5)
 => ([],1)
 => 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1000 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
00000 => [6] => ([],6)
 => ([],1)
 => 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
10000 => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001067
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
0111 => [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
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
1111 => [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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 0
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 0
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001276
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
1111 => [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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 0
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 0
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 0
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional. The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St001826
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001826: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => ([],1)
 => 0
1 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => ([],1)
 => 0
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
10 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
000 => [4] => ([],4)
 => ([],1)
 => 0
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
100 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
111 => [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)
 => 0
0000 => [5] => ([],5)
 => ([],1)
 => 0
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
1000 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
00000 => [6] => ([],6)
 => ([],1)
 => 0
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
10000 => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
Description
The maximal number of leaves on a vertex of a graph.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1 = 0 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000025The number of initial rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St001479The number of bridges of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000315The number of isolated vertices of a graph. St000352The Elizalde-Pak rank of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000260The radius of a connected graph. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000552The number of cut vertices of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001691The number of kings in a graph. St000917The open packing number of a graph. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian.