Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 16 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 1 = 2 - 1
00 => 00 => 0 = 1 - 1
01 => 10 => 1 = 2 - 1
10 => 01 => 0 = 1 - 1
11 => 11 => 2 = 3 - 1
000 => 000 => 0 = 1 - 1
001 => 100 => 1 = 2 - 1
010 => 010 => 0 = 1 - 1
011 => 110 => 2 = 3 - 1
100 => 001 => 0 = 1 - 1
101 => 101 => 1 = 2 - 1
110 => 011 => 0 = 1 - 1
111 => 111 => 3 = 4 - 1
0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 1 = 2 - 1
0010 => 0100 => 0 = 1 - 1
0011 => 1100 => 2 = 3 - 1
0100 => 0010 => 0 = 1 - 1
0101 => 1010 => 1 = 2 - 1
0110 => 0110 => 0 = 1 - 1
0111 => 1110 => 3 = 4 - 1
1000 => 0001 => 0 = 1 - 1
1001 => 1001 => 1 = 2 - 1
1010 => 0101 => 0 = 1 - 1
1011 => 1101 => 2 = 3 - 1
1100 => 0011 => 0 = 1 - 1
1101 => 1011 => 1 = 2 - 1
1110 => 0111 => 0 = 1 - 1
1111 => 1111 => 4 = 5 - 1
00000 => 00000 => 0 = 1 - 1
00001 => 10000 => 1 = 2 - 1
00010 => 01000 => 0 = 1 - 1
00011 => 11000 => 2 = 3 - 1
00100 => 00100 => 0 = 1 - 1
00101 => 10100 => 1 = 2 - 1
00110 => 01100 => 0 = 1 - 1
00111 => 11100 => 3 = 4 - 1
01000 => 00010 => 0 = 1 - 1
01001 => 10010 => 1 = 2 - 1
01010 => 01010 => 0 = 1 - 1
01011 => 11010 => 2 = 3 - 1
01100 => 00110 => 0 = 1 - 1
01101 => 10110 => 1 = 2 - 1
01110 => 01110 => 0 = 1 - 1
01111 => 11110 => 4 = 5 - 1
10000 => 00001 => 0 = 1 - 1
10001 => 10001 => 1 = 2 - 1
10010 => 01001 => 0 = 1 - 1
10011 => 11001 => 2 = 3 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 10 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => 1
1 => 1 => 0 => 2
00 => 00 => 11 => 1
01 => 10 => 01 => 2
10 => 01 => 10 => 1
11 => 11 => 00 => 3
000 => 000 => 111 => 1
001 => 100 => 011 => 2
010 => 010 => 101 => 1
011 => 110 => 001 => 3
100 => 001 => 110 => 1
101 => 101 => 010 => 2
110 => 011 => 100 => 1
111 => 111 => 000 => 4
0000 => 0000 => 1111 => 1
0001 => 1000 => 0111 => 2
0010 => 0100 => 1011 => 1
0011 => 1100 => 0011 => 3
0100 => 0010 => 1101 => 1
0101 => 1010 => 0101 => 2
0110 => 0110 => 1001 => 1
0111 => 1110 => 0001 => 4
1000 => 0001 => 1110 => 1
1001 => 1001 => 0110 => 2
1010 => 0101 => 1010 => 1
1011 => 1101 => 0010 => 3
1100 => 0011 => 1100 => 1
1101 => 1011 => 0100 => 2
1110 => 0111 => 1000 => 1
1111 => 1111 => 0000 => 5
00000 => 00000 => 11111 => 1
00001 => 10000 => 01111 => 2
00010 => 01000 => 10111 => 1
00011 => 11000 => 00111 => 3
00100 => 00100 => 11011 => 1
00101 => 10100 => 01011 => 2
00110 => 01100 => 10011 => 1
00111 => 11100 => 00011 => 4
01000 => 00010 => 11101 => 1
01001 => 10010 => 01101 => 2
01010 => 01010 => 10101 => 1
01011 => 11010 => 00101 => 3
01100 => 00110 => 11001 => 1
01101 => 10110 => 01001 => 2
01110 => 01110 => 10001 => 1
01111 => 11110 => 00001 => 5
10000 => 00001 => 11110 => 1
10001 => 10001 => 01110 => 2
10010 => 01001 => 10110 => 1
10011 => 11001 => 00110 => 3
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: St000382
(load all 7 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => 1
1 => [1,1] => [2] => 2
00 => [3] => [1,1,1] => 1
01 => [2,1] => [2,1] => 2
10 => [1,2] => [1,2] => 1
11 => [1,1,1] => [3] => 3
000 => [4] => [1,1,1,1] => 1
001 => [3,1] => [2,1,1] => 2
010 => [2,2] => [1,2,1] => 1
011 => [2,1,1] => [3,1] => 3
100 => [1,3] => [1,1,2] => 1
101 => [1,2,1] => [2,2] => 2
110 => [1,1,2] => [1,3] => 1
111 => [1,1,1,1] => [4] => 4
0000 => [5] => [1,1,1,1,1] => 1
0001 => [4,1] => [2,1,1,1] => 2
0010 => [3,2] => [1,2,1,1] => 1
0011 => [3,1,1] => [3,1,1] => 3
0100 => [2,3] => [1,1,2,1] => 1
0101 => [2,2,1] => [2,2,1] => 2
0110 => [2,1,2] => [1,3,1] => 1
0111 => [2,1,1,1] => [4,1] => 4
1000 => [1,4] => [1,1,1,2] => 1
1001 => [1,3,1] => [2,1,2] => 2
1010 => [1,2,2] => [1,2,2] => 1
1011 => [1,2,1,1] => [3,2] => 3
1100 => [1,1,3] => [1,1,3] => 1
1101 => [1,1,2,1] => [2,3] => 2
1110 => [1,1,1,2] => [1,4] => 1
1111 => [1,1,1,1,1] => [5] => 5
00000 => [6] => [1,1,1,1,1,1] => 1
00001 => [5,1] => [2,1,1,1,1] => 2
00010 => [4,2] => [1,2,1,1,1] => 1
00011 => [4,1,1] => [3,1,1,1] => 3
00100 => [3,3] => [1,1,2,1,1] => 1
00101 => [3,2,1] => [2,2,1,1] => 2
00110 => [3,1,2] => [1,3,1,1] => 1
00111 => [3,1,1,1] => [4,1,1] => 4
01000 => [2,4] => [1,1,1,2,1] => 1
01001 => [2,3,1] => [2,1,2,1] => 2
01010 => [2,2,2] => [1,2,2,1] => 1
01011 => [2,2,1,1] => [3,2,1] => 3
01100 => [2,1,3] => [1,1,3,1] => 1
01101 => [2,1,2,1] => [2,3,1] => 2
01110 => [2,1,1,2] => [1,4,1] => 1
01111 => [2,1,1,1,1] => [5,1] => 5
10000 => [1,5] => [1,1,1,1,2] => 1
10001 => [1,4,1] => [2,1,1,2] => 2
10010 => [1,3,2] => [1,2,1,2] => 1
10011 => [1,3,1,1] => [3,1,2] => 3
Description
The first part of an integer composition.
Matching statistic: St000678
(load all 4 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: St000439
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [2] => [1,1,0,0]
 => 3 = 2 + 1
00 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
01 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
10 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
001 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
010 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
100 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
101 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
0000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
0001 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
0010 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
0011 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
0100 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
1000 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1001 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
1100 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
00000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00001 => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
00010 => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00011 => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
00100 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
00101 => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 5 = 4 + 1
01000 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
01100 => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 5 + 1
10000 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
10001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
10010 => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
0100011 => [2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
0100111 => [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
0101011 => [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
0101111 => [2,2,1,1,1,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
0110011 => [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
0110111 => [2,1,2,1,1,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4 + 1
1000111 => [1,4,1,1,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 + 1
1001011 => [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
1001111 => [1,3,1,1,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5 + 1
1010011 => [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
1011011 => [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
1100011 => [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
1100111 => [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
1101011 => [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
1101111 => [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5 + 1
1110011 => [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
1111011 => [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000383
(load all 12 compositions to match this statistic)
Mapping
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 90%●distinct values known / distinct values provided: 88%
Values
0 => 0 => 1 => [1,1] => 1
1 => 1 => 0 => [2] => 2
00 => 00 => 01 => [2,1] => 1
01 => 01 => 10 => [1,2] => 2
10 => 10 => 11 => [1,1,1] => 1
11 => 11 => 00 => [3] => 3
000 => 000 => 001 => [3,1] => 1
001 => 001 => 010 => [2,2] => 2
010 => 100 => 011 => [2,1,1] => 1
011 => 011 => 100 => [1,3] => 3
100 => 010 => 101 => [1,2,1] => 1
101 => 101 => 110 => [1,1,2] => 2
110 => 110 => 111 => [1,1,1,1] => 1
111 => 111 => 000 => [4] => 4
0000 => 0000 => 0001 => [4,1] => 1
0001 => 0001 => 0010 => [3,2] => 2
0010 => 1000 => 0011 => [3,1,1] => 1
0011 => 0011 => 0100 => [2,3] => 3
0100 => 0100 => 1001 => [1,3,1] => 1
0101 => 1001 => 0110 => [2,1,2] => 2
0110 => 1100 => 0111 => [2,1,1,1] => 1
0111 => 0111 => 1000 => [1,4] => 4
1000 => 0010 => 0101 => [2,2,1] => 1
1001 => 0101 => 1010 => [1,2,2] => 2
1010 => 0110 => 1011 => [1,2,1,1] => 1
1011 => 1011 => 1100 => [1,1,3] => 3
1100 => 1010 => 1101 => [1,1,2,1] => 1
1101 => 1101 => 1110 => [1,1,1,2] => 2
1110 => 1110 => 1111 => [1,1,1,1,1] => 1
1111 => 1111 => 0000 => [5] => 5
00000 => 00000 => 00001 => [5,1] => 1
00001 => 00001 => 00010 => [4,2] => 2
00010 => 10000 => 00011 => [4,1,1] => 1
00011 => 00011 => 00100 => [3,3] => 3
00100 => 01000 => 10001 => [1,4,1] => 1
00101 => 10001 => 00110 => [3,1,2] => 2
00110 => 11000 => 00111 => [3,1,1,1] => 1
00111 => 00111 => 01000 => [2,4] => 4
01000 => 00100 => 01001 => [2,3,1] => 1
01001 => 01001 => 10010 => [1,3,2] => 2
01010 => 01100 => 10011 => [1,3,1,1] => 1
01011 => 10011 => 01100 => [2,1,3] => 3
01100 => 10100 => 11001 => [1,1,3,1] => 1
01101 => 11001 => 01110 => [2,1,1,2] => 2
01110 => 11100 => 01111 => [2,1,1,1,1] => 1
01111 => 01111 => 10000 => [1,5] => 5
10000 => 00010 => 00101 => [3,2,1] => 1
10001 => 00101 => 01010 => [2,2,2] => 2
10010 => 00110 => 01011 => [2,2,1,1] => 1
10011 => 01011 => 10100 => [1,2,3] => 3
0100111 => 0100111 => 1001000 => [1,3,4] => ? = 4
0101111 => 1001111 => 0110000 => [2,1,5] => ? = 5
0110111 => 1100111 => 0111000 => [2,1,1,4] => ? = 4
0111011 => 1110011 => 0111100 => [2,1,1,1,3] => ? = 3
0111110 => 1111100 => 0111111 => [2,1,1,1,1,1,1] => ? = 1
1000001 => 0000101 => 0001010 => [4,2,2] => ? = 2
1000010 => 0000110 => 0001011 => [4,2,1,1] => ? = 1
1000110 => 0001110 => 0010111 => [3,2,1,1,1] => ? = 1
1000111 => 0010111 => 0101000 => [2,2,4] => ? = 4
1001010 => 1000110 => 0011011 => [3,1,2,1,1] => ? = 1
1001100 => 1100010 => 0011101 => [3,1,1,2,1] => ? = 1
1001111 => 0101111 => 1010000 => [1,2,5] => ? = 5
1011011 => 0111011 => 1011100 => [1,2,1,1,3] => ? = 3
1011111 => 1011111 => 1100000 => [1,1,6] => ? = 6
1100111 => 1010111 => 1101000 => [1,1,2,4] => ? = 4
1101011 => 1011011 => 1101100 => [1,1,2,1,3] => ? = 3
1101111 => 1101111 => 1110000 => [1,1,1,5] => ? = 5
1110011 => 1101011 => 1110100 => [1,1,1,2,3] => ? = 3
1110111 => 1110111 => 1111000 => [1,1,1,1,4] => ? = 4
1111011 => 1111011 => 1111100 => [1,1,1,1,1,3] => ? = 3
1111101 => 1111101 => 1111110 => [1,1,1,1,1,1,2] => ? = 2
1111111 => 1111111 => 0000000 => [8] => ? = 8
Description
The last part of an integer composition.
Matching statistic: St000011
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 3
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 3
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
 => ? = 2
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 3
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
 => ? = 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 3
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 2
1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 3
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 2
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 1
1000111 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => ? = 2
1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 3
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 2
1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 2
1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 3
1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 1
1010111 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 2
1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
 => ? = 1
1011011 => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 3
1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 3
1100101 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 2
1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
 => ? = 2
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 1
1110001 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
1110101 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 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.
Matching statistic: St000993
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => [2]
 => 1
1 => [1,1] => [[1,1],[]]
 => [1,1]
 => 2
00 => [3] => [[3],[]]
 => [3]
 => 1
01 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 2
10 => [1,2] => [[2,1],[]]
 => [2,1]
 => 1
11 => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 3
000 => [4] => [[4],[]]
 => [4]
 => 1
001 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 2
010 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 3
100 => [1,3] => [[3,1],[]]
 => [3,1]
 => 1
101 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 2
110 => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 4
0000 => [5] => [[5],[]]
 => [5]
 => 1
0001 => [4,1] => [[4,4],[3]]
 => [4,4]
 => 2
0010 => [3,2] => [[4,3],[2]]
 => [4,3]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 3
0100 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 4
1000 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 3
1100 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 5
00000 => [6] => [[6],[]]
 => [6]
 => 1
00001 => [5,1] => [[5,5],[4]]
 => [5,5]
 => 2
00010 => [4,2] => [[5,4],[3]]
 => [5,4]
 => 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [4,4,4]
 => 3
00100 => [3,3] => [[5,3],[2]]
 => [5,3]
 => 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [4,4,3]
 => 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [3,3,3,3]
 => 4
01000 => [2,4] => [[5,2],[1]]
 => [5,2]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [4,4,2]
 => 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => 3
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => 5
10000 => [1,5] => [[5,1],[]]
 => [5,1]
 => 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => 2
10010 => [1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => 3
000011 => [5,1,1] => [[5,5,5],[4,4]]
 => [5,5,5]
 => ? = 3
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => [5,5,4]
 => ? = 2
000110 => [4,1,2] => [[5,4,4],[3,3]]
 => [5,4,4]
 => ? = 1
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [4,4,4,4]
 => ? = 4
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => [5,5,3]
 => ? = 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [4,4,4,3]
 => ? = 3
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => [4,4,3,3]
 => ? = 2
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [4,3,3,3]
 => ? = 1
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [3,3,3,3,3]
 => ? = 5
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? = 3
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [4,4,3,2]
 => ? = 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? = 4
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? = 3
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? = 3
100111 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? = 4
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 2
0100010 => [2,4,2] => [[6,5,2],[4,1]]
 => [6,5,2]
 => ? = 1
0100011 => [2,4,1,1] => [[5,5,5,2],[4,4,1]]
 => [5,5,5,2]
 => ? = 3
0100101 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
 => [5,5,4,2]
 => ? = 2
0100110 => [2,3,1,2] => [[5,4,4,2],[3,3,1]]
 => [5,4,4,2]
 => ? = 1
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => [4,4,4,4,2]
 => ? = 4
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 1
0101001 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
 => [5,5,3,2]
 => ? = 2
0101010 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
 => [5,4,3,2]
 => ? = 1
0101011 => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
 => [4,4,4,3,2]
 => ? = 3
0101100 => [2,2,1,3] => [[5,3,3,2],[2,2,1]]
 => [5,3,3,2]
 => ? = 1
0101101 => [2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]]
 => [4,4,3,3,2]
 => ? = 2
0101110 => [2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]]
 => [4,3,3,3,2]
 => ? = 1
0101111 => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => [3,3,3,3,3,2]
 => ? = 5
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ?
 => ? = 1
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 2
0110010 => [2,1,3,2] => [[5,4,2,2],[3,1,1]]
 => [5,4,2,2]
 => ? = 1
0110011 => [2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]]
 => [4,4,4,2,2]
 => ? = 3
0110101 => [2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]]
 => [4,4,3,2,2]
 => ? = 2
0110110 => [2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]]
 => [4,3,3,2,2]
 => ? = 1
0110111 => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => [3,3,3,3,2,2]
 => ? = 4
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? = 1
0111001 => [2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]]
 => [4,4,2,2,2]
 => ? = 2
0111010 => [2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]]
 => [4,3,2,2,2]
 => ? = 1
0111011 => [2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]]
 => [3,3,3,2,2,2]
 => ? = 3
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? = 1
0111101 => [2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]]
 => [3,3,2,2,2,2]
 => ? = 2
0111110 => [2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]]
 => [3,2,2,2,2,2]
 => ? = 1
0111111 => [2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
 => [2,2,2,2,2,2,2]
 => ? = 7
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 2
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 1
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? = 3
1000101 => [1,4,2,1] => [[5,5,4,1],[4,3]]
 => [5,5,4,1]
 => ? = 2
1000110 => [1,4,1,2] => [[5,4,4,1],[3,3]]
 => [5,4,4,1]
 => ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001050
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 88%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 1
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 2
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2,3,4},{5},{6}}
 => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6}}
 => 4
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => 3
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6},{7,8}}
 => ? = 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2},{3,4,5,6},{7},{8}}
 => ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5},{6,7,8}}
 => ? = 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4,5},{6,7},{8}}
 => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2},{3,4,5},{6},{7,8}}
 => ? = 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2},{3,4,5},{6},{7},{8}}
 => ? = 4
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4},{5,6,7},{8}}
 => ? = 2
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3,4},{5},{6,7,8}}
 => ? = 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5},{6,7},{8}}
 => ? = 2
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3,4},{5},{6},{7,8}}
 => ? = 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 5
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2},{3},{4,5,6,7},{8}}
 => ? = 2
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2},{3},{4,5,6},{7,8}}
 => ? = 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2},{3},{4,5,6},{7},{8}}
 => ? = 3
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5},{6,7,8}}
 => ? = 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6,7},{8}}
 => ? = 2
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5},{6},{7,8}}
 => ? = 1
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4,5},{6},{7},{8}}
 => ? = 4
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3},{4},{5,6,7,8}}
 => ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3},{4},{5,6,7},{8}}
 => ? = 2
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6},{7,8}}
 => ? = 1
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4},{5,6},{7},{8}}
 => ? = 3
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4},{5},{6,7,8}}
 => ? = 1
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4},{5},{6,7},{8}}
 => ? = 2
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 1
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6},{7},{8}}
 => ? = 7
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => {{1},{2,3,4,5,6},{7,8}}
 => ? = 1
1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => {{1},{2,3,4,5,6},{7},{8}}
 => ? = 3
1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => {{1},{2,3,4,5},{6,7,8}}
 => ? = 1
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => {{1},{2,3,4,5},{6,7},{8}}
 => ? = 2
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => {{1},{2,3,4,5},{6},{7,8}}
 => ? = 1
1000111 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => {{1},{2,3,4,5},{6},{7},{8}}
 => ? = 4
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4},{5,6,7,8}}
 => ? = 1
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 2
1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6},{7,8}}
 => ? = 1
1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3,4},{5,6},{7},{8}}
 => ? = 3
1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => {{1},{2,3,4},{5},{6,7,8}}
 => ? = 1
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => {{1},{2,3,4},{5},{6,7},{8}}
 => ? = 2
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => {{1},{2,3,4},{5},{6},{7,8}}
 => ? = 1
1001111 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6},{7},{8}}
 => ? = 5
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3},{4,5,6,7,8}}
 => ? = 1
1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3},{4,5,6,7},{8}}
 => ? = 2
1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3},{4,5,6},{7,8}}
 => ? = 1
1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3},{4,5,6},{7},{8}}
 => ? = 3
1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => {{1},{2,3},{4,5},{6,7,8}}
 => ? = 1
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => {{1},{2,3},{4,5},{6,7},{8}}
 => ? = 2
Description
The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St001135
(load all 3 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 88%
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
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 4
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 2
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1
1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 3
1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
1000111 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
1001111 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
The following 25 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. St000026The position of the first return of a Dyck path. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. 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. St000989The number of final rises of a permutation. St000260The radius of a connected graph. St000990The first ascent of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. 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. St000773The multiplicity of the largest Laplacian eigenvalue in a graph.