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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000390
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 01 => 1
1 => [1,1] => 11 => 11 => 1
00 => [3] => 100 => 001 => 1
01 => [2,1] => 101 => 101 => 2
10 => [1,2] => 110 => 011 => 1
11 => [1,1,1] => 111 => 111 => 1
000 => [4] => 1000 => 0001 => 1
001 => [3,1] => 1001 => 1001 => 2
010 => [2,2] => 1010 => 0101 => 2
011 => [2,1,1] => 1011 => 1101 => 2
100 => [1,3] => 1100 => 0011 => 1
101 => [1,2,1] => 1101 => 1011 => 2
110 => [1,1,2] => 1110 => 0111 => 1
111 => [1,1,1,1] => 1111 => 1111 => 1
0000 => [5] => 10000 => 00001 => 1
0001 => [4,1] => 10001 => 10001 => 2
0010 => [3,2] => 10010 => 01001 => 2
0011 => [3,1,1] => 10011 => 11001 => 2
0100 => [2,3] => 10100 => 00101 => 2
0101 => [2,2,1] => 10101 => 10101 => 3
0110 => [2,1,2] => 10110 => 01101 => 2
0111 => [2,1,1,1] => 10111 => 11101 => 2
1000 => [1,4] => 11000 => 00011 => 1
1001 => [1,3,1] => 11001 => 10011 => 2
1010 => [1,2,2] => 11010 => 01011 => 2
1011 => [1,2,1,1] => 11011 => 11011 => 2
1100 => [1,1,3] => 11100 => 00111 => 1
1101 => [1,1,2,1] => 11101 => 10111 => 2
1110 => [1,1,1,2] => 11110 => 01111 => 1
1111 => [1,1,1,1,1] => 11111 => 11111 => 1
00000 => [6] => 100000 => 000001 => 1
00001 => [5,1] => 100001 => 100001 => 2
00010 => [4,2] => 100010 => 010001 => 2
00011 => [4,1,1] => 100011 => 110001 => 2
00100 => [3,3] => 100100 => 001001 => 2
00101 => [3,2,1] => 100101 => 101001 => 3
00110 => [3,1,2] => 100110 => 011001 => 2
00111 => [3,1,1,1] => 100111 => 111001 => 2
01000 => [2,4] => 101000 => 000101 => 2
01001 => [2,3,1] => 101001 => 100101 => 3
01010 => [2,2,2] => 101010 => 010101 => 3
01011 => [2,2,1,1] => 101011 => 110101 => 3
01100 => [2,1,3] => 101100 => 001101 => 2
01101 => [2,1,2,1] => 101101 => 101101 => 3
01110 => [2,1,1,2] => 101110 => 011101 => 2
01111 => [2,1,1,1,1] => 101111 => 111101 => 2
10000 => [1,5] => 110000 => 000011 => 1
10001 => [1,4,1] => 110001 => 100011 => 2
10010 => [1,3,2] => 110010 => 010011 => 2
10011 => [1,3,1,1] => 110011 => 110011 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 01 => 0 = 1 - 1
1 => [1,1] => 11 => 11 => 0 = 1 - 1
00 => [3] => 100 => 001 => 0 = 1 - 1
01 => [2,1] => 101 => 101 => 1 = 2 - 1
10 => [1,2] => 110 => 011 => 0 = 1 - 1
11 => [1,1,1] => 111 => 111 => 0 = 1 - 1
000 => [4] => 1000 => 0001 => 0 = 1 - 1
001 => [3,1] => 1001 => 1001 => 1 = 2 - 1
010 => [2,2] => 1010 => 0101 => 1 = 2 - 1
011 => [2,1,1] => 1011 => 1101 => 1 = 2 - 1
100 => [1,3] => 1100 => 0011 => 0 = 1 - 1
101 => [1,2,1] => 1101 => 1011 => 1 = 2 - 1
110 => [1,1,2] => 1110 => 0111 => 0 = 1 - 1
111 => [1,1,1,1] => 1111 => 1111 => 0 = 1 - 1
0000 => [5] => 10000 => 00001 => 0 = 1 - 1
0001 => [4,1] => 10001 => 10001 => 1 = 2 - 1
0010 => [3,2] => 10010 => 01001 => 1 = 2 - 1
0011 => [3,1,1] => 10011 => 11001 => 1 = 2 - 1
0100 => [2,3] => 10100 => 00101 => 1 = 2 - 1
0101 => [2,2,1] => 10101 => 10101 => 2 = 3 - 1
0110 => [2,1,2] => 10110 => 01101 => 1 = 2 - 1
0111 => [2,1,1,1] => 10111 => 11101 => 1 = 2 - 1
1000 => [1,4] => 11000 => 00011 => 0 = 1 - 1
1001 => [1,3,1] => 11001 => 10011 => 1 = 2 - 1
1010 => [1,2,2] => 11010 => 01011 => 1 = 2 - 1
1011 => [1,2,1,1] => 11011 => 11011 => 1 = 2 - 1
1100 => [1,1,3] => 11100 => 00111 => 0 = 1 - 1
1101 => [1,1,2,1] => 11101 => 10111 => 1 = 2 - 1
1110 => [1,1,1,2] => 11110 => 01111 => 0 = 1 - 1
1111 => [1,1,1,1,1] => 11111 => 11111 => 0 = 1 - 1
00000 => [6] => 100000 => 000001 => 0 = 1 - 1
00001 => [5,1] => 100001 => 100001 => 1 = 2 - 1
00010 => [4,2] => 100010 => 010001 => 1 = 2 - 1
00011 => [4,1,1] => 100011 => 110001 => 1 = 2 - 1
00100 => [3,3] => 100100 => 001001 => 1 = 2 - 1
00101 => [3,2,1] => 100101 => 101001 => 2 = 3 - 1
00110 => [3,1,2] => 100110 => 011001 => 1 = 2 - 1
00111 => [3,1,1,1] => 100111 => 111001 => 1 = 2 - 1
01000 => [2,4] => 101000 => 000101 => 1 = 2 - 1
01001 => [2,3,1] => 101001 => 100101 => 2 = 3 - 1
01010 => [2,2,2] => 101010 => 010101 => 2 = 3 - 1
01011 => [2,2,1,1] => 101011 => 110101 => 2 = 3 - 1
01100 => [2,1,3] => 101100 => 001101 => 1 = 2 - 1
01101 => [2,1,2,1] => 101101 => 101101 => 2 = 3 - 1
01110 => [2,1,1,2] => 101110 => 011101 => 1 = 2 - 1
01111 => [2,1,1,1,1] => 101111 => 111101 => 1 = 2 - 1
10000 => [1,5] => 110000 => 000011 => 0 = 1 - 1
10001 => [1,4,1] => 110001 => 100011 => 1 = 2 - 1
10010 => [1,3,2] => 110010 => 010011 => 1 = 2 - 1
10011 => [1,3,1,1] => 110011 => 110011 => 1 = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St000292: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 0 = 1 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 1 = 2 - 1
100 => 0 = 1 - 1
101 => 1 = 2 - 1
110 => 0 = 1 - 1
111 => 0 = 1 - 1
0000 => 0 = 1 - 1
0001 => 1 = 2 - 1
0010 => 1 = 2 - 1
0011 => 1 = 2 - 1
0100 => 1 = 2 - 1
0101 => 2 = 3 - 1
0110 => 1 = 2 - 1
0111 => 1 = 2 - 1
1000 => 0 = 1 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 1 = 2 - 1
1100 => 0 = 1 - 1
1101 => 1 = 2 - 1
1110 => 0 = 1 - 1
1111 => 0 = 1 - 1
00000 => 0 = 1 - 1
00001 => 1 = 2 - 1
00010 => 1 = 2 - 1
00011 => 1 = 2 - 1
00100 => 1 = 2 - 1
00101 => 2 = 3 - 1
00110 => 1 = 2 - 1
00111 => 1 = 2 - 1
01000 => 1 = 2 - 1
01001 => 2 = 3 - 1
01010 => 2 = 3 - 1
01011 => 2 = 3 - 1
01100 => 1 = 2 - 1
01101 => 2 = 3 - 1
01110 => 1 = 2 - 1
01111 => 1 = 2 - 1
10000 => 0 = 1 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 1 = 2 - 1
=> ? = 1 - 1
11010101000 => ? = 4 - 1
11010100100 => ? = 4 - 1
11010010100 => ? = 4 - 1
11001010100 => ? = 4 - 1
10101010100 => ? = 5 - 1
00101010101 => ? = 6 - 1
00101010011 => ? = 5 - 1
00100101011 => ? = 5 - 1
0011111110 => ? = 2 - 1
0111111100 => ? = 2 - 1
01101010100 => ? = 5 - 1
01101001010 => ? = 5 - 1
01010110100 => ? = 5 - 1
11110001000 => ? = 2 - 1
11000111000 => ? = 2 - 1
00011100011 => ? = 3 - 1
00010001111 => ? = 3 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 67%
Values
0 => [2] => [1,1,0,0]
=> 0 = 1 - 1
1 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
01 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 2 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
00000111 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
00001001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
00001011 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
00001101 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
00001111 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
00010001 => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
00010010 => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
00010011 => [4,3,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
00010100 => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
00010101 => [4,2,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
00010110 => [4,2,1,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
00010111 => [4,2,1,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
00011000 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
00011001 => [4,1,3,1] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
00011010 => [4,1,2,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
00011011 => [4,1,2,1,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
00011100 => [4,1,1,3] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000159
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: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%
Values
0 => [2] => [[2],[]]
=> []
=> 0 = 1 - 1
1 => [1,1] => [[1,1],[]]
=> []
=> 0 = 1 - 1
00 => [3] => [[3],[]]
=> []
=> 0 = 1 - 1
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 1 = 2 - 1
10 => [1,2] => [[2,1],[]]
=> []
=> 0 = 1 - 1
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 0 = 1 - 1
000 => [4] => [[4],[]]
=> []
=> 0 = 1 - 1
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1 = 2 - 1
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 1 = 2 - 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
100 => [1,3] => [[3,1],[]]
=> []
=> 0 = 1 - 1
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 = 2 - 1
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 0 = 1 - 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0 = 1 - 1
0000 => [5] => [[5],[]]
=> []
=> 0 = 1 - 1
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 1 = 2 - 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 1 = 2 - 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2 = 3 - 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
1000 => [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1 = 2 - 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1 = 2 - 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 2 - 1
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 0 = 1 - 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 = 2 - 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0 = 1 - 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
00000 => [6] => [[6],[]]
=> []
=> 0 = 1 - 1
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1 = 2 - 1
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 1 = 2 - 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1 = 2 - 1
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 1 = 2 - 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2 = 3 - 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1 = 2 - 1
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 1 = 2 - 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2 = 3 - 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2 = 3 - 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2 = 3 - 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 2 - 1
10000 => [1,5] => [[5,1],[]]
=> []
=> 0 = 1 - 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1 = 2 - 1
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1 = 2 - 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1 = 2 - 1
0000010 => [6,2] => [[7,6],[5]]
=> ?
=> ? = 2 - 1
0000100 => [5,3] => [[7,5],[4]]
=> ?
=> ? = 2 - 1
0000101 => [5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? = 3 - 1
0000110 => [5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ? = 2 - 1
0010000 => [3,5] => [[7,3],[2]]
=> ?
=> ? = 2 - 1
0011000 => [3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ? = 2 - 1
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ? = 2 - 1
0100000 => [2,6] => [[7,2],[1]]
=> ?
=> ? = 2 - 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ? = 3 - 1
0101000 => [2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ? = 3 - 1
0110000 => [2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ? = 2 - 1
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ? = 3 - 1
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ?
=> ? = 2 - 1
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
=> ?
=> ? = 2 - 1
1000001 => [1,6,1] => [[6,6,1],[5]]
=> ?
=> ? = 2 - 1
1000010 => [1,5,2] => [[6,5,1],[4]]
=> ?
=> ? = 2 - 1
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
=> ?
=> ? = 2 - 1
1001000 => [1,3,4] => [[6,3,1],[2]]
=> ?
=> ? = 2 - 1
1001111 => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ? = 2 - 1
1010000 => [1,2,5] => [[6,2,1],[1]]
=> ?
=> ? = 2 - 1
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? = 3 - 1
1011000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2 - 1
1011100 => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? = 2 - 1
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2 - 1
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 - 1
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2 - 1
1101000 => [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2 - 1
1101100 => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2 - 1
1110001 => [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2 - 1
1110011 => [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2 - 1
1110100 => [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2 - 1
1111001 => [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2 - 1
00000010 => [7,2] => [[8,7],[6]]
=> ?
=> ? = 2 - 1
00000100 => [6,3] => [[8,6],[5]]
=> ?
=> ? = 2 - 1
00000101 => [6,2,1] => [[7,7,6],[6,5]]
=> ?
=> ? = 3 - 1
00000110 => [6,1,2] => [[7,6,6],[5,5]]
=> ?
=> ? = 2 - 1
00000111 => [6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> ? = 2 - 1
00001000 => [5,4] => [[8,5],[4]]
=> ?
=> ? = 2 - 1
00001001 => [5,3,1] => [[7,7,5],[6,4]]
=> ?
=> ? = 3 - 1
00001010 => [5,2,2] => [[7,6,5],[5,4]]
=> ?
=> ? = 3 - 1
00001011 => [5,2,1,1] => [[6,6,6,5],[5,5,4]]
=> ?
=> ? = 3 - 1
00001100 => [5,1,3] => [[7,5,5],[4,4]]
=> ?
=> ? = 2 - 1
00001101 => [5,1,2,1] => [[6,6,5,5],[5,4,4]]
=> ?
=> ? = 3 - 1
00001110 => [5,1,1,2] => [[6,5,5,5],[4,4,4]]
=> ?
=> ? = 2 - 1
00001111 => [5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> ? = 2 - 1
00010000 => [4,5] => [[8,4],[3]]
=> ?
=> ? = 2 - 1
00010001 => [4,4,1] => [[7,7,4],[6,3]]
=> ?
=> ? = 3 - 1
00010010 => [4,3,2] => [[7,6,4],[5,3]]
=> ?
=> ? = 3 - 1
00010011 => [4,3,1,1] => [[6,6,6,4],[5,5,3]]
=> ?
=> ? = 3 - 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
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: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Values
0 => [2] => [[2],[]]
=> []
=> 1
1 => [1,1] => [[1,1],[]]
=> []
=> 1
00 => [3] => [[3],[]]
=> []
=> 1
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 2
10 => [1,2] => [[2,1],[]]
=> []
=> 1
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 1
000 => [4] => [[4],[]]
=> []
=> 1
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
100 => [1,3] => [[3,1],[]]
=> []
=> 1
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 2
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 1
0000 => [5] => [[5],[]]
=> []
=> 1
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 2
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
1000 => [1,4] => [[4,1],[]]
=> []
=> 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 1
00000 => [6] => [[6],[]]
=> []
=> 1
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 2
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 3
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 2
10000 => [1,5] => [[5,1],[]]
=> []
=> 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
0000010 => [6,2] => [[7,6],[5]]
=> ?
=> ? = 2
0000100 => [5,3] => [[7,5],[4]]
=> ?
=> ? = 2
0000101 => [5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? = 3
0000110 => [5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ? = 2
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> ? = 2
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> ? = 2
0010000 => [3,5] => [[7,3],[2]]
=> ?
=> ? = 2
0011000 => [3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ? = 2
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ? = 2
0100000 => [2,6] => [[7,2],[1]]
=> ?
=> ? = 2
0100001 => [2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ? = 3
0101000 => [2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ? = 3
0110000 => [2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ? = 2
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ? = 3
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ?
=> ? = 2
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
=> ?
=> ? = 2
1000001 => [1,6,1] => [[6,6,1],[5]]
=> ?
=> ? = 2
1000010 => [1,5,2] => [[6,5,1],[4]]
=> ?
=> ? = 2
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
=> ?
=> ? = 2
1001000 => [1,3,4] => [[6,3,1],[2]]
=> ?
=> ? = 2
1001111 => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ? = 2
1010000 => [1,2,5] => [[6,2,1],[1]]
=> ?
=> ? = 2
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? = 3
1011000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2
1011100 => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? = 2
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2
1101000 => [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
1101100 => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2
1110001 => [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
1110011 => [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2
1110100 => [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2
1111001 => [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2
00000010 => [7,2] => [[8,7],[6]]
=> ?
=> ? = 2
00000011 => [7,1,1] => [[7,7,7],[6,6]]
=> [6,6]
=> ? = 2
00000100 => [6,3] => [[8,6],[5]]
=> ?
=> ? = 2
00000101 => [6,2,1] => [[7,7,6],[6,5]]
=> ?
=> ? = 3
00000110 => [6,1,2] => [[7,6,6],[5,5]]
=> ?
=> ? = 2
00000111 => [6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> ? = 2
00001000 => [5,4] => [[8,5],[4]]
=> ?
=> ? = 2
00001001 => [5,3,1] => [[7,7,5],[6,4]]
=> ?
=> ? = 3
00001010 => [5,2,2] => [[7,6,5],[5,4]]
=> ?
=> ? = 3
00001011 => [5,2,1,1] => [[6,6,6,5],[5,5,4]]
=> ?
=> ? = 3
00001100 => [5,1,3] => [[7,5,5],[4,4]]
=> ?
=> ? = 2
00001101 => [5,1,2,1] => [[6,6,5,5],[5,4,4]]
=> ?
=> ? = 3
00001110 => [5,1,1,2] => [[6,5,5,5],[4,4,4]]
=> ?
=> ? = 2
00001111 => [5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> ? = 2
00010000 => [4,5] => [[8,4],[3]]
=> ?
=> ? = 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001037
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 1
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 = 2 - 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 = 2 - 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 = 1 - 1
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 = 2 - 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 = 2 - 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 = 2 - 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 = 1 - 1
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 = 2 - 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 = 1 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
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 = 1 - 1
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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 1
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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
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 = 2 - 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 = 3 - 1
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 = 2 - 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 = 2 - 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 = 1 - 1
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 = 2 - 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 = 2 - 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 = 2 - 1
1101000 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
1101011 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
1101100 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
1101101 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
1101111 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
1110001 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
1110011 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
1110100 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
1110101 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
1111001 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
1111011 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
00000111 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
00001001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
00001011 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
00001101 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
00001111 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
00010001 => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
00010010 => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
00010011 => [4,3,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
00010100 => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
00010101 => [4,2,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
00010110 => [4,2,1,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ?
=> ? = 3 - 1
00010111 => [4,2,1,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3 - 1
00011000 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
00011001 => [4,1,3,1] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001124
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: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
=> []
=> ? = 1 - 2
1 => [1,1] => [[1,1],[]]
=> []
=> ? = 1 - 2
00 => [3] => [[3],[]]
=> []
=> ? = 1 - 2
01 => [2,1] => [[2,2],[1]]
=> [1]
=> ? = 2 - 2
10 => [1,2] => [[2,1],[]]
=> []
=> ? = 1 - 2
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? = 1 - 2
000 => [4] => [[4],[]]
=> []
=> ? = 1 - 2
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0 = 2 - 2
010 => [2,2] => [[3,2],[1]]
=> [1]
=> ? = 2 - 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
100 => [1,3] => [[3,1],[]]
=> []
=> ? = 1 - 2
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? = 2 - 2
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? = 1 - 2
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? = 1 - 2
0000 => [5] => [[5],[]]
=> []
=> ? = 1 - 2
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 0 = 2 - 2
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 0 = 2 - 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0 = 2 - 2
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> ? = 2 - 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1 = 3 - 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0 = 2 - 2
1000 => [1,4] => [[4,1],[]]
=> []
=> ? = 1 - 2
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0 = 2 - 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? = 2 - 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0 = 2 - 2
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? = 1 - 2
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? = 2 - 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? = 1 - 2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? = 1 - 2
00000 => [6] => [[6],[]]
=> []
=> ? = 1 - 2
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 0 = 2 - 2
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 0 = 2 - 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0 = 2 - 2
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 0 = 2 - 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1 = 3 - 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0 = 2 - 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0 = 2 - 2
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> ? = 2 - 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1 = 3 - 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1 = 3 - 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1 = 3 - 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1 = 3 - 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0 = 2 - 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0 = 2 - 2
10000 => [1,5] => [[5,1],[]]
=> []
=> ? = 1 - 2
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0 = 2 - 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0 = 2 - 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0 = 2 - 2
10100 => [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? = 2 - 2
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1 = 3 - 2
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0 = 2 - 2
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0 = 2 - 2
11000 => [1,1,4] => [[4,1,1],[]]
=> []
=> ? = 1 - 2
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0 = 2 - 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? = 2 - 2
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0 = 2 - 2
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? = 1 - 2
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? = 2 - 2
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? = 1 - 2
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? = 1 - 2
000000 => [7] => [[7],[]]
=> []
=> ? = 1 - 2
000001 => [6,1] => [[6,6],[5]]
=> [5]
=> 0 = 2 - 2
000010 => [5,2] => [[6,5],[4]]
=> [4]
=> 0 = 2 - 2
000011 => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0 = 2 - 2
000100 => [4,3] => [[6,4],[3]]
=> [3]
=> 0 = 2 - 2
000101 => [4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> 1 = 3 - 2
000110 => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0 = 2 - 2
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 0 = 2 - 2
001000 => [3,4] => [[6,3],[2]]
=> [2]
=> 0 = 2 - 2
001001 => [3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> 1 = 3 - 2
001010 => [3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> 1 = 3 - 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> 1 = 3 - 2
001100 => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0 = 2 - 2
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> 1 = 3 - 2
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0 = 2 - 2
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 0 = 2 - 2
010000 => [2,5] => [[6,2],[1]]
=> [1]
=> ? = 2 - 2
010001 => [2,4,1] => [[5,5,2],[4,1]]
=> [4,1]
=> 1 = 3 - 2
010010 => [2,3,2] => [[5,4,2],[3,1]]
=> [3,1]
=> 1 = 3 - 2
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> 1 = 3 - 2
100000 => [1,6] => [[6,1],[]]
=> []
=> ? = 1 - 2
101000 => [1,2,4] => [[5,2,1],[1]]
=> [1]
=> ? = 2 - 2
110000 => [1,1,5] => [[5,1,1],[]]
=> []
=> ? = 1 - 2
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> ? = 2 - 2
111000 => [1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? = 1 - 2
111010 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> ? = 2 - 2
111100 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? = 1 - 2
111101 => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> ? = 2 - 2
111110 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? = 1 - 2
111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? = 1 - 2
0000000 => [8] => [[8],[]]
=> []
=> ? = 1 - 2
0000010 => [6,2] => [[7,6],[5]]
=> ?
=> ? = 2 - 2
0000100 => [5,3] => [[7,5],[4]]
=> ?
=> ? = 2 - 2
0000101 => [5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? = 3 - 2
0000110 => [5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ? = 2 - 2
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> ? = 2 - 2
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> ? = 2 - 2
0010000 => [3,5] => [[7,3],[2]]
=> ?
=> ? = 2 - 2
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: St000068
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
0 => [2] => [[2],[]]
=> ([(0,1)],2)
=> 1
1 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1
00 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
01 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
11 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
010 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
100 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
110 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
0010 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
0100 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
1000 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
00010 => [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
00100 => [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
01000 => [2,4] => [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
10000 => [1,5] => [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2
0000001 => [7,1] => [[7,7],[6]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
0000010 => [6,2] => [[7,6],[5]]
=> ?
=> ? = 2
0000011 => [6,1,1] => [[6,6,6],[5,5]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ? = 2
0000100 => [5,3] => [[7,5],[4]]
=> ?
=> ? = 2
0000101 => [5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? = 3
0000110 => [5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ? = 2
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 2
0001000 => [4,4] => [[7,4],[3]]
=> ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
=> ? = 2
0001001 => [4,3,1] => [[6,6,4],[5,3]]
=> ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
=> ? = 3
0001010 => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
=> ? = 3
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
=> ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
=> ? = 3
0001100 => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)
=> ? = 2
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
=> ? = 3
0001110 => [4,1,1,2] => [[5,4,4,4],[3,3,3]]
=> ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
=> ? = 2
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 2
0010000 => [3,5] => [[7,3],[2]]
=> ?
=> ? = 2
0010001 => [3,4,1] => [[6,6,3],[5,2]]
=> ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
=> ? = 3
0010010 => [3,3,2] => [[6,5,3],[4,2]]
=> ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)
=> ? = 3
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]]
=> ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
=> ? = 3
0010100 => [3,2,3] => [[6,4,3],[3,2]]
=> ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
=> ? = 3
0010101 => [3,2,2,1] => [[5,5,4,3],[4,3,2]]
=> ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
=> ? = 4
0010110 => [3,2,1,2] => [[5,4,4,3],[3,3,2]]
=> ([(0,6),(0,7),(1,5),(2,3),(2,4),(4,6),(5,7)],8)
=> ? = 3
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
=> ? = 3
0011000 => [3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ? = 2
0011001 => [3,1,3,1] => [[5,5,3,3],[4,2,2]]
=> ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
=> ? = 3
0011010 => [3,1,2,2] => [[5,4,3,3],[3,2,2]]
=> ([(0,5),(1,4),(1,6),(2,3),(2,6),(4,7),(5,7)],8)
=> ? = 3
0011011 => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
=> ? = 3
0011100 => [3,1,1,3] => [[5,3,3,3],[2,2,2]]
=> ([(0,4),(1,5),(1,6),(3,7),(4,7),(5,2),(6,3)],8)
=> ? = 2
0011101 => [3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
=> ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
=> ? = 3
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ? = 2
0011111 => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ? = 2
0100000 => [2,6] => [[7,2],[1]]
=> ?
=> ? = 2
0100001 => [2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ? = 3
0100010 => [2,4,2] => [[6,5,2],[4,1]]
=> ([(0,6),(1,4),(1,6),(2,3),(2,7),(4,5),(5,7)],8)
=> ? = 3
0100011 => [2,4,1,1] => [[5,5,5,2],[4,4,1]]
=> ([(0,6),(1,3),(2,4),(2,6),(3,7),(4,5),(5,7)],8)
=> ? = 3
0100100 => [2,3,3] => [[6,4,2],[3,1]]
=> ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
=> ? = 3
0100101 => [2,3,2,1] => [[5,5,4,2],[4,3,1]]
=> ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ? = 4
0100110 => [2,3,1,2] => [[5,4,4,2],[3,3,1]]
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(4,7),(5,7)],8)
=> ? = 3
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
=> ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
=> ? = 3
0101000 => [2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ? = 3
0101001 => [2,2,3,1] => [[5,5,3,2],[4,2,1]]
=> ([(0,7),(1,5),(2,5),(2,6),(3,4),(3,6),(4,7)],8)
=> ? = 4
0101010 => [2,2,2,2] => [[5,4,3,2],[3,2,1]]
=> ([(0,6),(1,5),(1,6),(2,5),(2,7),(3,4),(3,7)],8)
=> ? = 4
0101011 => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
=> ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
=> ? = 4
0101100 => [2,2,1,3] => [[5,3,3,2],[2,2,1]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
=> ? = 3
0101101 => [2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]]
=> ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8)
=> ? = 4
0101110 => [2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(4,5),(5,7)],8)
=> ? = 3
0101111 => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
=> ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
=> ? = 3
0110000 => [2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ? = 2
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ? = 3
0110010 => [2,1,3,2] => [[5,4,2,2],[3,1,1]]
=> ([(0,6),(1,3),(1,7),(2,4),(2,5),(4,6),(5,7)],8)
=> ? = 3
Description
The number of minimal elements in a poset.
Matching statistic: St000201
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: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
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: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [[.,.],.]
=> 1
1 => [1,1] => [1,0,1,0]
=> [.,[.,.]]
=> 1
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]
=> [.,[.,[.,.]]]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 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]
=> [.,[.,[.,[.,.]]]]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
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]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> 3
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 3
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]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 2
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 3
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 2
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 2
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 3
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 3
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> ? = 2
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 3
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 2
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 2
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 2
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 3
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 3
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 3
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> ? = 3
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 4
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 3
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> ? = 2
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 3
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
=> ? = 3
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> ? = 2
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> ? = 3
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ? = 2
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 2
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> ? = 3
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[.,.],[[[[.,.],.],.],[.,[.,.]]]]
=> ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],[[[.,.],.],.]]]
=> ? = 3
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> ? = 4
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[.,.],[[[.,.],.],[.,[[.,.],.]]]]
=> ? = 3
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[.,.],[[[.,.],.],[.,[.,[.,.]]]]]
=> ? = 3
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> ? = 4
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> ? = 4
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
=> ? = 3
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 4
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ? = 3
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 3
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ? = 2
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[.,.],[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 3
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 3
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]
=> [[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> ? = 3
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 4
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 3
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 3
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 2
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 3
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 3
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].
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000356The number of occurrences of the pattern 13-2. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000568The hook number of a binary tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. 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.
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