Your data matches 84 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000326
(load all 5 compositions to match this statistic)
Mapping
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 2
1 => 1
00 => 3
01 => 2
10 => 1
11 => 1
000 => 4
001 => 3
010 => 2
011 => 2
100 => 1
101 => 1
110 => 1
111 => 1
0000 => 5
0001 => 4
0010 => 3
0011 => 3
0100 => 2
0101 => 2
0110 => 2
0111 => 2
1000 => 1
1001 => 1
1010 => 1
1011 => 1
1100 => 1
1101 => 1
1110 => 1
1111 => 1
00000 => 6
00001 => 5
00010 => 4
00011 => 4
00100 => 3
00101 => 3
00110 => 3
00111 => 3
01000 => 2
01001 => 2
01010 => 2
01011 => 2
01100 => 2
01101 => 2
01110 => 2
01111 => 2
10000 => 1
10001 => 1
10010 => 1
10011 => 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: St000297
(load all 3 compositions to match this statistic)
Mapping
St000297: 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 => 0 = 1 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 0 = 1 - 1
010 => 0 = 1 - 1
011 => 0 = 1 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 1
0000 => 0 = 1 - 1
0001 => 0 = 1 - 1
0010 => 0 = 1 - 1
0011 => 0 = 1 - 1
0100 => 0 = 1 - 1
0101 => 0 = 1 - 1
0110 => 0 = 1 - 1
0111 => 0 = 1 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 1 = 2 - 1
1100 => 2 = 3 - 1
1101 => 2 = 3 - 1
1110 => 3 = 4 - 1
1111 => 4 = 5 - 1
00000 => 0 = 1 - 1
00001 => 0 = 1 - 1
00010 => 0 = 1 - 1
00011 => 0 = 1 - 1
00100 => 0 = 1 - 1
00101 => 0 = 1 - 1
00110 => 0 = 1 - 1
00111 => 0 = 1 - 1
01000 => 0 = 1 - 1
01001 => 0 = 1 - 1
01010 => 0 = 1 - 1
01011 => 0 = 1 - 1
01100 => 0 = 1 - 1
01101 => 0 = 1 - 1
01110 => 0 = 1 - 1
01111 => 0 = 1 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 1 = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000382
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: 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] => 1
11 => [1,1,1] => 1
000 => [4] => 4
001 => [3,1] => 3
010 => [2,2] => 2
011 => [2,1,1] => 2
100 => [1,3] => 1
101 => [1,2,1] => 1
110 => [1,1,2] => 1
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] => 2
0101 => [2,2,1] => 2
0110 => [2,1,2] => 2
0111 => [2,1,1,1] => 2
1000 => [1,4] => 1
1001 => [1,3,1] => 1
1010 => [1,2,2] => 1
1011 => [1,2,1,1] => 1
1100 => [1,1,3] => 1
1101 => [1,1,2,1] => 1
1110 => [1,1,1,2] => 1
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] => 2
01001 => [2,3,1] => 2
01010 => [2,2,2] => 2
01011 => [2,2,1,1] => 2
01100 => [2,1,3] => 2
01101 => [2,1,2,1] => 2
01110 => [2,1,1,2] => 2
01111 => [2,1,1,1,1] => 2
10000 => [1,5] => 1
10001 => [1,4,1] => 1
10010 => [1,3,2] => 1
10011 => [1,3,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: 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] => 1
10 => [1,2] => 2
11 => [1,1,1] => 1
000 => [4] => 4
001 => [3,1] => 1
010 => [2,2] => 2
011 => [2,1,1] => 1
100 => [1,3] => 3
101 => [1,2,1] => 1
110 => [1,1,2] => 2
111 => [1,1,1,1] => 1
0000 => [5] => 5
0001 => [4,1] => 1
0010 => [3,2] => 2
0011 => [3,1,1] => 1
0100 => [2,3] => 3
0101 => [2,2,1] => 1
0110 => [2,1,2] => 2
0111 => [2,1,1,1] => 1
1000 => [1,4] => 4
1001 => [1,3,1] => 1
1010 => [1,2,2] => 2
1011 => [1,2,1,1] => 1
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 1
1110 => [1,1,1,2] => 2
1111 => [1,1,1,1,1] => 1
00000 => [6] => 6
00001 => [5,1] => 1
00010 => [4,2] => 2
00011 => [4,1,1] => 1
00100 => [3,3] => 3
00101 => [3,2,1] => 1
00110 => [3,1,2] => 2
00111 => [3,1,1,1] => 1
01000 => [2,4] => 4
01001 => [2,3,1] => 1
01010 => [2,2,2] => 2
01011 => [2,2,1,1] => 1
01100 => [2,1,3] => 3
01101 => [2,1,2,1] => 1
01110 => [2,1,1,2] => 2
01111 => [2,1,1,1,1] => 1
10000 => [1,5] => 5
10001 => [1,4,1] => 1
10010 => [1,3,2] => 2
10011 => [1,3,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000025
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: 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]
 => 1
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]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
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]
 => 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]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
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]
 => 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]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 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]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: 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]
 => 1
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]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
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]
 => 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]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
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]
 => 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]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 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]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000678
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: 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]
 => 1
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]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001135
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: 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]
 => 1
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]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 3 = 2 + 1
1 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
00 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 5 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 5 = 4 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 5 = 4 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000010
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => [1,1]
 => 2
1 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
00 => [3] => ([],3)
 => [1,1,1]
 => 3
01 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
10 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
000 => [4] => ([],4)
 => [1,1,1,1]
 => 4
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
100 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 3
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 5
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 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)
 => [5]
 => 1
1000 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 4
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 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)
 => [5]
 => 1
00000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 6
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 2
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 3
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
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]
 => 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 4
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
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]
 => 1
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3
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]
 => 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)
 => [5,1]
 => 2
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]
 => 1
10000 => [1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 5
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
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]
 => 1
Description
The length of the partition.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000759The smallest missing part in an integer partition. St000823The number of unsplittable factors of the set partition. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001733The number of weak left to right maxima of a Dyck path. St001176The size of a partition minus its first part. St001657The number of twos in an integer partition. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St000286The number of connected components of the complement of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000160The multiplicity of the smallest part of a partition. St000993The multiplicity of the largest part of an integer partition. St000617The number of global maxima of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000989The number of final rises of a permutation. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000054The first entry of the permutation. St000007The number of saliances of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000738The first entry in the last row of a standard tableau. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000287The number of connected components of a graph. 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. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000553The number of blocks of a graph. St000991The number of right-to-left minima of a permutation. St001202Call 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St000475The number of parts equal to 1 in a partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001557The number of inversions of the second entry of a permutation. St001330The hat guessing number of a graph.