Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000292
(load all 14 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 0
1 => [1,1] => 11 => 0
00 => [3] => 100 => 0
01 => [2,1] => 101 => 1
10 => [1,2] => 110 => 0
11 => [1,1,1] => 111 => 0
000 => [4] => 1000 => 0
001 => [3,1] => 1001 => 1
010 => [2,2] => 1010 => 1
011 => [2,1,1] => 1011 => 1
100 => [1,3] => 1100 => 0
101 => [1,2,1] => 1101 => 1
110 => [1,1,2] => 1110 => 0
111 => [1,1,1,1] => 1111 => 0
0000 => [5] => 10000 => 0
0001 => [4,1] => 10001 => 1
0010 => [3,2] => 10010 => 1
0011 => [3,1,1] => 10011 => 1
0100 => [2,3] => 10100 => 1
0101 => [2,2,1] => 10101 => 2
0110 => [2,1,2] => 10110 => 1
0111 => [2,1,1,1] => 10111 => 1
1000 => [1,4] => 11000 => 0
1001 => [1,3,1] => 11001 => 1
1010 => [1,2,2] => 11010 => 1
1011 => [1,2,1,1] => 11011 => 1
1100 => [1,1,3] => 11100 => 0
1101 => [1,1,2,1] => 11101 => 1
1110 => [1,1,1,2] => 11110 => 0
1111 => [1,1,1,1,1] => 11111 => 0
00000 => [6] => 100000 => 0
00001 => [5,1] => 100001 => 1
00010 => [4,2] => 100010 => 1
00011 => [4,1,1] => 100011 => 1
00100 => [3,3] => 100100 => 1
00101 => [3,2,1] => 100101 => 2
00110 => [3,1,2] => 100110 => 1
00111 => [3,1,1,1] => 100111 => 1
01000 => [2,4] => 101000 => 1
01001 => [2,3,1] => 101001 => 2
01010 => [2,2,2] => 101010 => 2
01011 => [2,2,1,1] => 101011 => 2
01100 => [2,1,3] => 101100 => 1
01101 => [2,1,2,1] => 101101 => 2
01110 => [2,1,1,2] => 101110 => 1
01111 => [2,1,1,1,1] => 101111 => 1
10000 => [1,5] => 110000 => 0
10001 => [1,4,1] => 110001 => 1
10010 => [1,3,2] => 110010 => 1
10011 => [1,3,1,1] => 110011 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 1 = 0 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 2 = 1 + 1
10 => [1,2] => 110 => 1 = 0 + 1
11 => [1,1,1] => 111 => 1 = 0 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 2 = 1 + 1
010 => [2,2] => 1010 => 2 = 1 + 1
011 => [2,1,1] => 1011 => 2 = 1 + 1
100 => [1,3] => 1100 => 1 = 0 + 1
101 => [1,2,1] => 1101 => 2 = 1 + 1
110 => [1,1,2] => 1110 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 1 = 0 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 2 = 1 + 1
0010 => [3,2] => 10010 => 2 = 1 + 1
0011 => [3,1,1] => 10011 => 2 = 1 + 1
0100 => [2,3] => 10100 => 2 = 1 + 1
0101 => [2,2,1] => 10101 => 3 = 2 + 1
0110 => [2,1,2] => 10110 => 2 = 1 + 1
0111 => [2,1,1,1] => 10111 => 2 = 1 + 1
1000 => [1,4] => 11000 => 1 = 0 + 1
1001 => [1,3,1] => 11001 => 2 = 1 + 1
1010 => [1,2,2] => 11010 => 2 = 1 + 1
1011 => [1,2,1,1] => 11011 => 2 = 1 + 1
1100 => [1,1,3] => 11100 => 1 = 0 + 1
1101 => [1,1,2,1] => 11101 => 2 = 1 + 1
1110 => [1,1,1,2] => 11110 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 1 = 0 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 2 = 1 + 1
00010 => [4,2] => 100010 => 2 = 1 + 1
00011 => [4,1,1] => 100011 => 2 = 1 + 1
00100 => [3,3] => 100100 => 2 = 1 + 1
00101 => [3,2,1] => 100101 => 3 = 2 + 1
00110 => [3,1,2] => 100110 => 2 = 1 + 1
00111 => [3,1,1,1] => 100111 => 2 = 1 + 1
01000 => [2,4] => 101000 => 2 = 1 + 1
01001 => [2,3,1] => 101001 => 3 = 2 + 1
01010 => [2,2,2] => 101010 => 3 = 2 + 1
01011 => [2,2,1,1] => 101011 => 3 = 2 + 1
01100 => [2,1,3] => 101100 => 2 = 1 + 1
01101 => [2,1,2,1] => 101101 => 3 = 2 + 1
01110 => [2,1,1,2] => 101110 => 2 = 1 + 1
01111 => [2,1,1,1,1] => 101111 => 2 = 1 + 1
10000 => [1,5] => 110000 => 1 = 0 + 1
10001 => [1,4,1] => 110001 => 2 = 1 + 1
10010 => [1,3,2] => 110010 => 2 = 1 + 1
10011 => [1,3,1,1] => 110011 => 2 = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 17 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 11 => 0
1 => [1,1] => 11 => 00 => 0
00 => [3] => 100 => 011 => 0
01 => [2,1] => 101 => 110 => 1
10 => [1,2] => 110 => 111 => 0
11 => [1,1,1] => 111 => 000 => 0
000 => [4] => 1000 => 0011 => 0
001 => [3,1] => 1001 => 0110 => 1
010 => [2,2] => 1010 => 1101 => 1
011 => [2,1,1] => 1011 => 1100 => 1
100 => [1,3] => 1100 => 0111 => 0
101 => [1,2,1] => 1101 => 1110 => 1
110 => [1,1,2] => 1110 => 1111 => 0
111 => [1,1,1,1] => 1111 => 0000 => 0
0000 => [5] => 10000 => 00011 => 0
0001 => [4,1] => 10001 => 00110 => 1
0010 => [3,2] => 10010 => 01101 => 1
0011 => [3,1,1] => 10011 => 01100 => 1
0100 => [2,3] => 10100 => 11001 => 1
0101 => [2,2,1] => 10101 => 11010 => 2
0110 => [2,1,2] => 10110 => 11011 => 1
0111 => [2,1,1,1] => 10111 => 11000 => 1
1000 => [1,4] => 11000 => 00111 => 0
1001 => [1,3,1] => 11001 => 01110 => 1
1010 => [1,2,2] => 11010 => 11101 => 1
1011 => [1,2,1,1] => 11011 => 11100 => 1
1100 => [1,1,3] => 11100 => 01111 => 0
1101 => [1,1,2,1] => 11101 => 11110 => 1
1110 => [1,1,1,2] => 11110 => 11111 => 0
1111 => [1,1,1,1,1] => 11111 => 00000 => 0
00000 => [6] => 100000 => 000011 => 0
00001 => [5,1] => 100001 => 000110 => 1
00010 => [4,2] => 100010 => 001101 => 1
00011 => [4,1,1] => 100011 => 001100 => 1
00100 => [3,3] => 100100 => 011001 => 1
00101 => [3,2,1] => 100101 => 011010 => 2
00110 => [3,1,2] => 100110 => 011011 => 1
00111 => [3,1,1,1] => 100111 => 011000 => 1
01000 => [2,4] => 101000 => 110001 => 1
01001 => [2,3,1] => 101001 => 110010 => 2
01010 => [2,2,2] => 101010 => 110101 => 2
01011 => [2,2,1,1] => 101011 => 110100 => 2
01100 => [2,1,3] => 101100 => 110011 => 1
01101 => [2,1,2,1] => 101101 => 110110 => 2
01110 => [2,1,1,2] => 101110 => 110111 => 1
01111 => [2,1,1,1,1] => 101111 => 110000 => 1
10000 => [1,5] => 110000 => 000111 => 0
10001 => [1,4,1] => 110001 => 001110 => 1
10010 => [1,3,2] => 110010 => 011101 => 1
10011 => [1,3,1,1] => 110011 => 011100 => 1
Description
The number of descents of a binary word.
Matching statistic: St001037
(load all 7 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000386
(load all 8 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
St000386: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => [1,0,1,0]
 => 0
1 => [1,1] => [2] => [1,1,0,0]
 => 0
00 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 0
11 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
010 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
100 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
101 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
0010 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
0011 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
0100 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
1000 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 0
1001 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
1100 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
00000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
00001 => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
00010 => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
00011 => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
00100 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
00101 => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
00110 => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
00111 => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
01000 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
01001 => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1
01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
10000 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
10001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
10010 => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
10011 => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
0000111 => [5,1,1,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
0001011 => [4,2,1,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
0001111 => [4,1,1,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
0010011 => [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
0010111 => [3,2,1,1,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
0011011 => [3,1,2,1,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
0011111 => [3,1,1,1,1,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 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]
 => ? = 2
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]
 => ? = 2
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
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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 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]
 => ? = 2
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]
 => ? = 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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 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]
 => ? = 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000159
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => []
 => 0
1 => [1,1] => [[1,1],[]]
 => []
 => 0
00 => [3] => [[3],[]]
 => []
 => 0
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 1
10 => [1,2] => [[2,1],[]]
 => []
 => 0
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => 0
000 => [4] => [[4],[]]
 => []
 => 0
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
100 => [1,3] => [[3,1],[]]
 => []
 => 0
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 0
0000 => [5] => [[5],[]]
 => []
 => 0
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
1000 => [1,4] => [[4,1],[]]
 => []
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 0
00000 => [6] => [[6],[]]
 => []
 => 0
00001 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 1
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 1
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
10000 => [1,5] => [[5,1],[]]
 => []
 => 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 1
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 1
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 2
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 1
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 1
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? = 1
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? = 1
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 2
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 2
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ?
 => ? = 1
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 2
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? = 1
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? = 1
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 1
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 1
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? = 1
1001000 => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 1
1001111 => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? = 1
1010000 => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
 => ?
 => ? = 2
1011000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 1
1011100 => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
 => ?
 => ? = 1
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
 => ?
 => ? = 1
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
 => ?
 => ? = 1
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 1
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => []
 => 1 = 0 + 1
1 => [1,1] => [[1,1],[]]
 => []
 => 1 = 0 + 1
00 => [3] => [[3],[]]
 => []
 => 1 = 0 + 1
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 2 = 1 + 1
10 => [1,2] => [[2,1],[]]
 => []
 => 1 = 0 + 1
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => 1 = 0 + 1
000 => [4] => [[4],[]]
 => []
 => 1 = 0 + 1
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 2 = 1 + 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 2 = 1 + 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2 = 1 + 1
100 => [1,3] => [[3,1],[]]
 => []
 => 1 = 0 + 1
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 2 = 1 + 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => 1 = 0 + 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 1 = 0 + 1
0000 => [5] => [[5],[]]
 => []
 => 1 = 0 + 1
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 2 = 1 + 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 2 = 1 + 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2 = 1 + 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 2 = 1 + 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 3 = 2 + 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 2 = 1 + 1
1000 => [1,4] => [[4,1],[]]
 => []
 => 1 = 0 + 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2 = 1 + 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 2 = 1 + 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => 1 = 0 + 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 1 = 0 + 1
00000 => [6] => [[6],[]]
 => []
 => 1 = 0 + 1
00001 => [5,1] => [[5,5],[4]]
 => [4]
 => 2 = 1 + 1
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 2 = 1 + 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2 = 1 + 1
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 2 = 1 + 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 3 = 2 + 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2 = 1 + 1
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => 2 = 1 + 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 3 = 2 + 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 3 = 2 + 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 3 = 2 + 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 3 = 2 + 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 2 = 1 + 1
10000 => [1,5] => [[5,1],[]]
 => []
 => 1 = 0 + 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 2 = 1 + 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 2 = 1 + 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2 = 1 + 1
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 1 + 1
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 1 + 1
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 2 + 1
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 1 + 1
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [4,4,4]
 => ? = 1 + 1
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [3,3,3,3]
 => ? = 1 + 1
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 1 + 1
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? = 1 + 1
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? = 1 + 1
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1 + 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 2 + 1
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 2 + 1
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ?
 => ? = 1 + 1
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 2 + 1
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? = 1 + 1
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? = 1 + 1
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 1 + 1
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 1 + 1
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? = 1 + 1
1001000 => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 1 + 1
1001111 => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? = 1 + 1
1010000 => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1 + 1
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
 => ?
 => ? = 2 + 1
1011000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 1 + 1
1011100 => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
 => ?
 => ? = 1 + 1
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
 => ?
 => ? = 1 + 1
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
 => ?
 => ? = 1 + 1
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 1 + 1
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 1 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001124
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 75%
Values
0 => [2] => [[2],[]]
 => []
 => ? = 0 - 1
1 => [1,1] => [[1,1],[]]
 => []
 => ? = 0 - 1
00 => [3] => [[3],[]]
 => []
 => ? = 0 - 1
01 => [2,1] => [[2,2],[1]]
 => [1]
 => ? = 1 - 1
10 => [1,2] => [[2,1],[]]
 => []
 => ? = 0 - 1
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? = 0 - 1
000 => [4] => [[4],[]]
 => []
 => ? = 0 - 1
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 0 = 1 - 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => ? = 1 - 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
100 => [1,3] => [[3,1],[]]
 => []
 => ? = 0 - 1
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? = 1 - 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? = 0 - 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? = 0 - 1
0000 => [5] => [[5],[]]
 => []
 => ? = 0 - 1
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 0 = 1 - 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 0 = 1 - 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0 = 1 - 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 1 - 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
1000 => [1,4] => [[4,1],[]]
 => []
 => ? = 0 - 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0 = 1 - 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? = 1 - 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? = 0 - 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? = 1 - 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? = 0 - 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? = 0 - 1
00000 => [6] => [[6],[]]
 => []
 => ? = 0 - 1
00001 => [5,1] => [[5,5],[4]]
 => [4]
 => 0 = 1 - 1
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 0 = 1 - 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0 = 1 - 1
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 0 = 1 - 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1 = 2 - 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 0 = 1 - 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 0 = 1 - 1
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1 - 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 1 = 2 - 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 1 = 2 - 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1 = 2 - 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0 = 1 - 1
10000 => [1,5] => [[5,1],[]]
 => []
 => ? = 0 - 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 0 = 1 - 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0 = 1 - 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 0 = 1 - 1
10100 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? = 1 - 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1 = 2 - 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
11000 => [1,1,4] => [[4,1,1],[]]
 => []
 => ? = 0 - 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0 = 1 - 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? = 1 - 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? = 0 - 1
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => ? = 1 - 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 0 - 1
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ? = 0 - 1
000000 => [7] => [[7],[]]
 => []
 => ? = 0 - 1
000001 => [6,1] => [[6,6],[5]]
 => [5]
 => 0 = 1 - 1
000010 => [5,2] => [[6,5],[4]]
 => [4]
 => 0 = 1 - 1
000011 => [5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => 0 = 1 - 1
000100 => [4,3] => [[6,4],[3]]
 => [3]
 => 0 = 1 - 1
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => [4,3]
 => 1 = 2 - 1
000110 => [4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 0 = 1 - 1
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => 0 = 1 - 1
001000 => [3,4] => [[6,3],[2]]
 => [2]
 => 0 = 1 - 1
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => [4,2]
 => 1 = 2 - 1
001010 => [3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => 1 = 2 - 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [3,3,2]
 => 1 = 2 - 1
001100 => [3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 0 = 1 - 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => [3,2,2]
 => 1 = 2 - 1
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => 0 = 1 - 1
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => 0 = 1 - 1
010000 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1 - 1
010001 => [2,4,1] => [[5,5,2],[4,1]]
 => [4,1]
 => 1 = 2 - 1
010010 => [2,3,2] => [[5,4,2],[3,1]]
 => [3,1]
 => 1 = 2 - 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [3,3,1]
 => 1 = 2 - 1
100000 => [1,6] => [[6,1],[]]
 => []
 => ? = 0 - 1
101000 => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => ? = 1 - 1
110000 => [1,1,5] => [[5,1,1],[]]
 => []
 => ? = 0 - 1
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => ? = 1 - 1
111000 => [1,1,1,4] => [[4,1,1,1],[]]
 => []
 => ? = 0 - 1
111010 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => ? = 1 - 1
111100 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 0 - 1
111101 => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => ? = 1 - 1
111110 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 0 - 1
111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => ? = 0 - 1
0000000 => [8] => [[8],[]]
 => []
 => ? = 0 - 1
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 1 - 1
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 1 - 1
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 2 - 1
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 1 - 1
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [4,4,4]
 => ? = 1 - 1
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [3,3,3,3]
 => ? = 1 - 1
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 1 - 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000201
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 1 = 0 + 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]
 => [.,[.,[[.,.],.]]]
 => 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 1 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 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]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 1 = 0 + 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]
 => [.,[.,[[[.,.],.],.]]]
 => 1 = 0 + 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]
 => [.,[.,[.,[[.,.],.]]]]
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 2 = 1 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => 2 = 1 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[.,.],.],[[.,.],[.,.]]]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => 2 = 1 + 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]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 2 = 1 + 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]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 1 = 0 + 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
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => ? = 1 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [[[[[.,.],.],.],.],[[[.,.],.],.]]
 => ? = 1 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
 => ? = 2 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
 => ? = 1 + 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => ? = 1 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
 => ? = 2 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
 => ? = 2 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
 => ? = 1 + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
 => ? = 2 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
 => ? = 1 + 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => ? = 1 + 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],.],[[[[[.,.],.],.],.],.]]
 => ? = 1 + 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
 => ? = 2 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
 => ? = 2 + 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
 => ? = 2 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [[[.,.],.],[[.,.],[[[.,.],.],.]]]
 => ? = 2 + 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
 => ? = 3 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
 => ? = 2 + 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
 => ? = 2 + 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => ? = 1 + 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
 => ? = 2 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
 => ? = 2 + 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
 => ? = 2 + 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => ? = 1 + 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
 => ? = 2 + 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
 => ? = 1 + 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => ? = 1 + 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
 => ? = 2 + 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [[.,.],[[[[.,.],.],.],[.,[.,.]]]]
 => ? = 2 + 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],[[[.,.],.],.]]]
 => ? = 2 + 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
 => ? = 3 + 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => ? = 2 + 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [[.,.],[[[.,.],.],[.,[.,[.,.]]]]]
 => ? = 2 + 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
 => ? = 3 + 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => ? = 3 + 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => ? = 2 + 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => ? = 3 + 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
 => ? = 2 + 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => ? = 2 + 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[[[[[.,.],.],.],.],.]]]
 => ? = 1 + 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [[.,.],[.,[[[[.,.],.],.],[.,.]]]]
 => ? = 2 + 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
 => ? = 2 + 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
 => ? = 2 + 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => ? = 2 + 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
 => ? = 3 + 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
 => ? = 2 + 1
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
 => ? = 2 + 1
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
 => ? = 1 + 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
 => ? = 2 + 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
 => ? = 2 + 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St001036
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 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]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 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]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
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]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 2
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 2
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 3
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 2
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 2
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 2
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 2
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 2
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 2
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 3
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 2
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 3
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 3
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 2
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 3
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 2
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 2
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 2
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 2
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 2
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
 => ? = 2
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St000568The hook number of a binary tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000356The number of occurrences of the pattern 13-2. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St000092The number of outer peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001487The number of inner corners of a skew partition. St000354The number of recoils of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.