Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000392
(load all 3 compositions to match this statistic)
Mapping
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 1 = 2 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 2 = 3 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 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 => 2 = 3 - 1
0111 => 3 = 4 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 2 = 3 - 1
1100 => 2 = 3 - 1
1101 => 2 = 3 - 1
1110 => 3 = 4 - 1
1111 => 4 = 5 - 1
00000 => 0 = 1 - 1
00001 => 1 = 2 - 1
00010 => 1 = 2 - 1
00011 => 2 = 3 - 1
00100 => 1 = 2 - 1
00101 => 1 = 2 - 1
00110 => 2 = 3 - 1
00111 => 3 = 4 - 1
01000 => 1 = 2 - 1
01001 => 1 = 2 - 1
01010 => 1 = 2 - 1
01011 => 2 = 3 - 1
01100 => 2 = 3 - 1
01101 => 2 = 3 - 1
01110 => 3 = 4 - 1
01111 => 4 = 5 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 2 = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 2
1 => [1,1] => 1
00 => [3] => 3
01 => [2,1] => 2
10 => [1,2] => 2
11 => [1,1,1] => 1
000 => [4] => 4
001 => [3,1] => 3
010 => [2,2] => 2
011 => [2,1,1] => 2
100 => [1,3] => 3
101 => [1,2,1] => 2
110 => [1,1,2] => 2
111 => [1,1,1,1] => 1
0000 => [5] => 5
0001 => [4,1] => 4
0010 => [3,2] => 3
0011 => [3,1,1] => 3
0100 => [2,3] => 3
0101 => [2,2,1] => 2
0110 => [2,1,2] => 2
0111 => [2,1,1,1] => 2
1000 => [1,4] => 4
1001 => [1,3,1] => 3
1010 => [1,2,2] => 2
1011 => [1,2,1,1] => 2
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 2
1110 => [1,1,1,2] => 2
1111 => [1,1,1,1,1] => 1
00000 => [6] => 6
00001 => [5,1] => 5
00010 => [4,2] => 4
00011 => [4,1,1] => 4
00100 => [3,3] => 3
00101 => [3,2,1] => 3
00110 => [3,1,2] => 3
00111 => [3,1,1,1] => 3
01000 => [2,4] => 4
01001 => [2,3,1] => 3
01010 => [2,2,2] => 2
01011 => [2,2,1,1] => 2
01100 => [2,1,3] => 3
01101 => [2,1,2,1] => 2
01110 => [2,1,1,2] => 2
01111 => [2,1,1,1,1] => 2
10000 => [1,5] => 5
10001 => [1,4,1] => 4
10010 => [1,3,2] => 3
10011 => [1,3,1,1] => 3
Description
The largest part of an integer composition.
Matching statistic: St000013
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 2
1 => [1,1] => [1,0,1,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => 3
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]
 => 4
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
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]
 => 3
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]
 => 5
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
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]
 => 4
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
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]
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000147
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 2
1 => [1,1] => [1,1]
 => 1
00 => [3] => [3]
 => 3
01 => [2,1] => [2,1]
 => 2
10 => [1,2] => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => 1
000 => [4] => [4]
 => 4
001 => [3,1] => [3,1]
 => 3
010 => [2,2] => [2,2]
 => 2
011 => [2,1,1] => [2,1,1]
 => 2
100 => [1,3] => [3,1]
 => 3
101 => [1,2,1] => [2,1,1]
 => 2
110 => [1,1,2] => [2,1,1]
 => 2
111 => [1,1,1,1] => [1,1,1,1]
 => 1
0000 => [5] => [5]
 => 5
0001 => [4,1] => [4,1]
 => 4
0010 => [3,2] => [3,2]
 => 3
0011 => [3,1,1] => [3,1,1]
 => 3
0100 => [2,3] => [3,2]
 => 3
0101 => [2,2,1] => [2,2,1]
 => 2
0110 => [2,1,2] => [2,2,1]
 => 2
0111 => [2,1,1,1] => [2,1,1,1]
 => 2
1000 => [1,4] => [4,1]
 => 4
1001 => [1,3,1] => [3,1,1]
 => 3
1010 => [1,2,2] => [2,2,1]
 => 2
1011 => [1,2,1,1] => [2,1,1,1]
 => 2
1100 => [1,1,3] => [3,1,1]
 => 3
1101 => [1,1,2,1] => [2,1,1,1]
 => 2
1110 => [1,1,1,2] => [2,1,1,1]
 => 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
00000 => [6] => [6]
 => 6
00001 => [5,1] => [5,1]
 => 5
00010 => [4,2] => [4,2]
 => 4
00011 => [4,1,1] => [4,1,1]
 => 4
00100 => [3,3] => [3,3]
 => 3
00101 => [3,2,1] => [3,2,1]
 => 3
00110 => [3,1,2] => [3,2,1]
 => 3
00111 => [3,1,1,1] => [3,1,1,1]
 => 3
01000 => [2,4] => [4,2]
 => 4
01001 => [2,3,1] => [3,2,1]
 => 3
01010 => [2,2,2] => [2,2,2]
 => 2
01011 => [2,2,1,1] => [2,2,1,1]
 => 2
01100 => [2,1,3] => [3,2,1]
 => 3
01101 => [2,1,2,1] => [2,2,1,1]
 => 2
01110 => [2,1,1,2] => [2,2,1,1]
 => 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 2
10000 => [1,5] => [5,1]
 => 5
10001 => [1,4,1] => [4,1,1]
 => 4
10010 => [1,3,2] => [3,2,1]
 => 3
10011 => [1,3,1,1] => [3,1,1,1]
 => 3
Description
The largest part of an integer partition.
Matching statistic: St000444
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 2
1 => [1,1] => [1,0,1,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => 3
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]
 => 4
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
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]
 => 3
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]
 => 5
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
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]
 => 4
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
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]
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000684
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: 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]
 => 2
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]
 => 3
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]
 => 3
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]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
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]
 => 3
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]
 => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
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]
 => 3
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]
 => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$. The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$. One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$. Examples: * For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Matching statistic: St000686
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000686: 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]
 => 2
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]
 => 3
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]
 => 3
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]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
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]
 => 3
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]
 => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
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]
 => 3
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]
 => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The finitistic dominant dimension of a Dyck path. To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
Matching statistic: St000982
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1
1 => [1,1] => 11 => 2
00 => [3] => 100 => 2
01 => [2,1] => 101 => 1
10 => [1,2] => 110 => 2
11 => [1,1,1] => 111 => 3
000 => [4] => 1000 => 3
001 => [3,1] => 1001 => 2
010 => [2,2] => 1010 => 1
011 => [2,1,1] => 1011 => 2
100 => [1,3] => 1100 => 2
101 => [1,2,1] => 1101 => 2
110 => [1,1,2] => 1110 => 3
111 => [1,1,1,1] => 1111 => 4
0000 => [5] => 10000 => 4
0001 => [4,1] => 10001 => 3
0010 => [3,2] => 10010 => 2
0011 => [3,1,1] => 10011 => 2
0100 => [2,3] => 10100 => 2
0101 => [2,2,1] => 10101 => 1
0110 => [2,1,2] => 10110 => 2
0111 => [2,1,1,1] => 10111 => 3
1000 => [1,4] => 11000 => 3
1001 => [1,3,1] => 11001 => 2
1010 => [1,2,2] => 11010 => 2
1011 => [1,2,1,1] => 11011 => 2
1100 => [1,1,3] => 11100 => 3
1101 => [1,1,2,1] => 11101 => 3
1110 => [1,1,1,2] => 11110 => 4
1111 => [1,1,1,1,1] => 11111 => 5
00000 => [6] => 100000 => 5
00001 => [5,1] => 100001 => 4
00010 => [4,2] => 100010 => 3
00011 => [4,1,1] => 100011 => 3
00100 => [3,3] => 100100 => 2
00101 => [3,2,1] => 100101 => 2
00110 => [3,1,2] => 100110 => 2
00111 => [3,1,1,1] => 100111 => 3
01000 => [2,4] => 101000 => 3
01001 => [2,3,1] => 101001 => 2
01010 => [2,2,2] => 101010 => 1
01011 => [2,2,1,1] => 101011 => 2
01100 => [2,1,3] => 101100 => 2
01101 => [2,1,2,1] => 101101 => 2
01110 => [2,1,1,2] => 101110 => 3
01111 => [2,1,1,1,1] => 101111 => 4
10000 => [1,5] => 110000 => 4
10001 => [1,4,1] => 110001 => 3
10010 => [1,3,2] => 110010 => 2
10011 => [1,3,1,1] => 110011 => 2
Description
The length of the longest constant subword.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 2
1 => [1,1] => 11 => 1
00 => [3] => 100 => 2
01 => [2,1] => 101 => 3
10 => [1,2] => 110 => 2
11 => [1,1,1] => 111 => 1
000 => [4] => 1000 => 2
001 => [3,1] => 1001 => 2
010 => [2,2] => 1010 => 4
011 => [2,1,1] => 1011 => 3
100 => [1,3] => 1100 => 2
101 => [1,2,1] => 1101 => 3
110 => [1,1,2] => 1110 => 2
111 => [1,1,1,1] => 1111 => 1
0000 => [5] => 10000 => 2
0001 => [4,1] => 10001 => 2
0010 => [3,2] => 10010 => 3
0011 => [3,1,1] => 10011 => 2
0100 => [2,3] => 10100 => 4
0101 => [2,2,1] => 10101 => 5
0110 => [2,1,2] => 10110 => 3
0111 => [2,1,1,1] => 10111 => 3
1000 => [1,4] => 11000 => 2
1001 => [1,3,1] => 11001 => 2
1010 => [1,2,2] => 11010 => 4
1011 => [1,2,1,1] => 11011 => 3
1100 => [1,1,3] => 11100 => 2
1101 => [1,1,2,1] => 11101 => 3
1110 => [1,1,1,2] => 11110 => 2
1111 => [1,1,1,1,1] => 11111 => 1
00000 => [6] => 100000 => 2
00001 => [5,1] => 100001 => 2
00010 => [4,2] => 100010 => 3
00011 => [4,1,1] => 100011 => 2
00100 => [3,3] => 100100 => 3
00101 => [3,2,1] => 100101 => 4
00110 => [3,1,2] => 100110 => 2
00111 => [3,1,1,1] => 100111 => 2
01000 => [2,4] => 101000 => 4
01001 => [2,3,1] => 101001 => 4
01010 => [2,2,2] => 101010 => 6
01011 => [2,2,1,1] => 101011 => 5
01100 => [2,1,3] => 101100 => 3
01101 => [2,1,2,1] => 101101 => 3
01110 => [2,1,1,2] => 101110 => 3
01111 => [2,1,1,1,1] => 101111 => 3
10000 => [1,5] => 110000 => 2
10001 => [1,4,1] => 110001 => 2
10010 => [1,3,2] => 110010 => 3
10011 => [1,3,1,1] => 110011 => 2
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000442
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: 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 = 2 - 1
1 => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
 => 2 = 3 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5 = 6 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4 = 5 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 3 = 4 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 4 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
Description
The maximal area to the right of an up step of a Dyck path.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. 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: St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St000503The maximal difference between two elements in a common block. St000521The number of distinct subtrees of an ordered tree. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000306The bounce count of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000662The staircase size of the code of a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St000028The number of stack-sorts needed to sort a permutation. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. 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. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St000094The depth of an ordered tree. St000454The largest eigenvalue of a graph if it is integral. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001645The pebbling number of a connected graph. St001060The distinguishing index of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.