Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000293
(load all 4 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 10 => 1
10 => 01 => 0
11 => 11 => 0
000 => 000 => 0
001 => 100 => 2
010 => 010 => 1
011 => 110 => 2
100 => 001 => 0
101 => 101 => 1
110 => 011 => 0
111 => 111 => 0
0000 => 0000 => 0
0001 => 1000 => 3
0010 => 0100 => 2
0011 => 1100 => 4
0100 => 0010 => 1
0101 => 1010 => 3
0110 => 0110 => 2
0111 => 1110 => 3
1000 => 0001 => 0
1001 => 1001 => 2
1010 => 0101 => 1
1011 => 1101 => 2
1100 => 0011 => 0
1101 => 1011 => 1
1110 => 0111 => 0
1111 => 1111 => 0
00000 => 00000 => 0
00001 => 10000 => 4
00010 => 01000 => 3
00011 => 11000 => 6
00100 => 00100 => 2
00101 => 10100 => 5
00110 => 01100 => 4
00111 => 11100 => 6
01000 => 00010 => 1
01001 => 10010 => 4
01010 => 01010 => 3
01011 => 11010 => 5
01100 => 00110 => 2
01101 => 10110 => 4
01110 => 01110 => 3
01111 => 11110 => 4
10000 => 00001 => 0
10001 => 10001 => 3
10010 => 01001 => 2
10011 => 11001 => 4
Description
The number of inversions of a binary word.
Matching statistic: St000290
(load all 2 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 00 => 0
01 => 10 => 10 => 1
10 => 01 => 01 => 0
11 => 11 => 11 => 0
000 => 000 => 000 => 0
001 => 100 => 010 => 2
010 => 010 => 100 => 1
011 => 110 => 110 => 2
100 => 001 => 001 => 0
101 => 101 => 101 => 1
110 => 011 => 011 => 0
111 => 111 => 111 => 0
0000 => 0000 => 0000 => 0
0001 => 1000 => 0010 => 3
0010 => 0100 => 0100 => 2
0011 => 1100 => 1010 => 4
0100 => 0010 => 1000 => 1
0101 => 1010 => 0110 => 3
0110 => 0110 => 1100 => 2
0111 => 1110 => 1110 => 3
1000 => 0001 => 0001 => 0
1001 => 1001 => 0101 => 2
1010 => 0101 => 1001 => 1
1011 => 1101 => 1101 => 2
1100 => 0011 => 0011 => 0
1101 => 1011 => 1011 => 1
1110 => 0111 => 0111 => 0
1111 => 1111 => 1111 => 0
00000 => 00000 => 00000 => 0
00001 => 10000 => 00010 => 4
00010 => 01000 => 00100 => 3
00011 => 11000 => 01010 => 6
00100 => 00100 => 01000 => 2
00101 => 10100 => 10010 => 5
00110 => 01100 => 10100 => 4
00111 => 11100 => 11010 => 6
01000 => 00010 => 10000 => 1
01001 => 10010 => 00110 => 4
01010 => 01010 => 01100 => 3
01011 => 11010 => 10110 => 5
01100 => 00110 => 11000 => 2
01101 => 10110 => 01110 => 4
01110 => 01110 => 11100 => 3
01111 => 11110 => 11110 => 4
10000 => 00001 => 00001 => 0
10001 => 10001 => 00101 => 3
10010 => 01001 => 01001 => 2
10011 => 11001 => 10101 => 4
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000369
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000369: 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]
 => 2
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]
 => 2
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]
 => 3
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
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]
 => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
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]
 => 3
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]
 => 2
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]
 => 2
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]
 => 4
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]
 => 3
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]
 => 6
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]
 => 2
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]
 => 5
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]
 => 4
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]
 => 6
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]
 => 4
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]
 => 3
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]
 => 5
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]
 => 2
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]
 => 4
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]
 => 3
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]
 => 4
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]
 => 3
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]
 => 2
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]
 => 4
Description
The dinv deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$ In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$. See also [[St000376]] for the bounce deficit.
Matching statistic: St000228
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 86%●distinct values known / distinct values provided: 85%
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]
 => 2
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2
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]
 => 3
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 4
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
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]
 => 3
1000 => [1,4] => [[4,1],[]]
 => []
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 2
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]
 => 4
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 3
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 6
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 5
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 4
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 6
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 4
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]
 => 5
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]
 => 4
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 4
10000 => [1,5] => [[5,1],[]]
 => []
 => 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 3
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 4
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 5
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 4
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 9
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 8
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [4,4,4]
 => ? = 12
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => [4,4,3]
 => ? = 11
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [3,3,3,3]
 => ? = 12
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 2
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => [3,3,3,2]
 => ? = 11
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? = 4
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? = 8
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 6
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]]
 => ?
 => ? = 6
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? = 3
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? = 4
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 5
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 4
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? = 8
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]]
 => ?
 => ? = 8
1010000 => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
 => ?
 => ? = 5
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]]
 => ?
 => ? = 3
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
 => ?
 => ? = 4
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
 => ?
 => ? = 3
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 6
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 6
1101000 => [1,1,2,4] => [[5,2,1,1],[1]]
 => ?
 => ? = 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St001175
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 84%●distinct values known / distinct values provided: 77%
Values
0 => [2] => [[2],[]]
 => [2]
 => 0
1 => [1,1] => [[1,1],[]]
 => [1,1]
 => 0
00 => [3] => [[3],[]]
 => [3]
 => 0
01 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 1
10 => [1,2] => [[2,1],[]]
 => [2,1]
 => 0
11 => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 0
000 => [4] => [[4],[]]
 => [4]
 => 0
001 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 2
010 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 2
100 => [1,3] => [[3,1],[]]
 => [3,1]
 => 0
101 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 0
0000 => [5] => [[5],[]]
 => [5]
 => 0
0001 => [4,1] => [[4,4],[3]]
 => [4,4]
 => 3
0010 => [3,2] => [[4,3],[2]]
 => [4,3]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 4
0100 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 3
1000 => [1,4] => [[4,1],[]]
 => [4,1]
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 2
1100 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 0
00000 => [6] => [[6],[]]
 => [6]
 => 0
00001 => [5,1] => [[5,5],[4]]
 => [5,5]
 => 4
00010 => [4,2] => [[5,4],[3]]
 => [5,4]
 => 3
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [4,4,4]
 => 6
00100 => [3,3] => [[5,3],[2]]
 => [5,3]
 => 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [4,4,3]
 => 5
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 4
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [3,3,3,3]
 => 6
01000 => [2,4] => [[5,2],[1]]
 => [5,2]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [4,4,2]
 => 4
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => 5
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => 4
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 3
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => 4
10000 => [1,5] => [[5,1],[]]
 => [5,1]
 => 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => 3
10010 => [1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => 4
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 5
0000011 => [6,1,1] => [[6,6,6],[5,5]]
 => [6,6,6]
 => ? = 10
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 4
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 9
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 8
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [5,5,5,5]
 => ? = 12
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => [5,5,5,4]
 => ? = 11
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => [5,5,4,4]
 => ? = 10
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [4,4,4,4,4]
 => ? = 12
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 2
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]]
 => [5,5,5,3]
 => ? = 10
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => [4,4,4,4,3]
 => ? = 11
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? = 4
0011011 => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
 => [4,4,4,3,3]
 => ? = 10
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? = 8
0011111 => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => [3,3,3,3,3,3]
 => ? = 10
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 6
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => [4,4,4,4,2]
 => ? = 10
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]]
 => ?
 => ? = 6
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? = 3
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? = 4
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 5
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 4
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? = 8
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]]
 => ?
 => ? = 8
1010000 => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
 => ?
 => ? = 5
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]]
 => ?
 => ? = 3
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
 => ?
 => ? = 4
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
 => ?
 => ? = 3
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 6
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 6
1101000 => [1,1,2,4] => [[5,2,1,1],[1]]
 => ?
 => ? = 1
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000747
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000747: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 77%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 0
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 3
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 4
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 4
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 3
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2,3,4},{5},{6}}
 => 6
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 5
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6}}
 => 6
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 5
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 4
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => 4
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 10
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 4
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 9
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 8
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 12
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 3
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 8
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 7
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 11
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 6
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4},{5},{6,7},{8}}
 => ? = 10
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 9
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 12
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3},{4,5,6,7,8}}
 => ? = 2
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3},{4,5,6,7},{8}}
 => ? = 7
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 6
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 10
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5},{6,7,8}}
 => ? = 5
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 9
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4,5},{6},{7,8}}
 => ? = 8
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4,5},{6},{7},{8}}
 => ? = 11
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 4
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 8
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => {{1,2,3},{4},{5,6},{7,8}}
 => ? = 7
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2,3},{4},{5,6},{7},{8}}
 => ? = 10
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 6
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2,3},{4},{5},{6,7},{8}}
 => ? = 9
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5},{6},{7,8}}
 => ? = 8
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6},{7},{8}}
 => ? = 10
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 6
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6},{7,8}}
 => ? = 5
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2},{3,4,5,6},{7},{8}}
 => ? = 9
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5},{6,7,8}}
 => ? = 4
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4,5},{6,7},{8}}
 => ? = 8
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2},{3,4,5},{6},{7,8}}
 => ? = 7
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2},{3,4,5},{6},{7},{8}}
 => ? = 10
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 3
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4},{5,6,7},{8}}
 => ? = 7
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 6
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 9
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3,4},{5},{6,7,8}}
 => ? = 5
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5},{6,7},{8}}
 => ? = 8
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3,4},{5},{6},{7,8}}
 => ? = 7
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 9
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 2
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2},{3},{4,5,6,7},{8}}
 => ? = 6
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2},{3},{4,5,6},{7,8}}
 => ? = 5
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2},{3},{4,5,6},{7},{8}}
 => ? = 8
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5},{6,7,8}}
 => ? = 4
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6,7},{8}}
 => ? = 7
Description
A variant of the major index of a set partition. For a set partition $P = B_1|\dots|B_k$ in canonical form (this is, each block is ordered increasingly and all blocks are ordered by their smallest element), one defined $\pi = \pi(P)$ to be the permutation obtained by writing the letters in all blocks as one-line notation and $\omega = \omega(P) = (\omega_1,\ldots,\omega_k)$ be to be the integer composition of the ordered block sizes. This statistic is then given in [1, (2.7)] by $$\operatorname{maj}(\pi) + \sum_{max\ B_i < min\ B_{i+1}} (\omega_1 + \cdots + \omega_i - i).$$
Matching statistic: St001438
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 38%
Values
0 => [2] => [[2],[]]
 => 0
1 => [1,1] => [[1,1],[]]
 => 0
00 => [3] => [[3],[]]
 => 0
01 => [2,1] => [[2,2],[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
010 => [2,2] => [[3,2],[1]]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => 2
100 => [1,3] => [[3,1],[]]
 => 0
101 => [1,2,1] => [[2,2,1],[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
0010 => [3,2] => [[4,3],[2]]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => 4
0100 => [2,3] => [[4,2],[1]]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => 3
1000 => [1,4] => [[4,1],[]]
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 2
1100 => [1,1,3] => [[3,1,1],[]]
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[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
00010 => [4,2] => [[5,4],[3]]
 => ? = 3
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 6
00100 => [3,3] => [[5,3],[2]]
 => ? = 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ? = 4
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ? = 6
01000 => [2,4] => [[5,2],[1]]
 => ? = 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 4
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ? = 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ? = 5
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ? = 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 4
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ? = 3
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ? = 4
10000 => [1,5] => [[5,1],[]]
 => ? = 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => ? = 3
10010 => [1,3,2] => [[4,3,1],[2]]
 => ? = 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ? = 4
10100 => [1,2,3] => [[4,2,1],[1]]
 => ? = 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 3
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ? = 2
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ? = 3
11000 => [1,1,4] => [[4,1,1],[]]
 => ? = 0
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ? = 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ? = 2
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => ? = 0
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ? = 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ? = 0
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ? = 0
000000 => [7] => [[7],[]]
 => ? = 0
000001 => [6,1] => [[6,6],[5]]
 => ? = 5
000010 => [5,2] => [[6,5],[4]]
 => ? = 4
000011 => [5,1,1] => [[5,5,5],[4,4]]
 => ? = 8
000100 => [4,3] => [[6,4],[3]]
 => ? = 3
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => ? = 7
000110 => [4,1,2] => [[5,4,4],[3,3]]
 => ? = 6
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => ? = 9
001000 => [3,4] => [[6,3],[2]]
 => ? = 2
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => ? = 6
001010 => [3,2,2] => [[5,4,3],[3,2]]
 => ? = 5
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => ? = 8
001100 => [3,1,3] => [[5,3,3],[2,2]]
 => ? = 4
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ? = 7
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => ? = 6
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => ? = 8
010000 => [2,5] => [[6,2],[1]]
 => ? = 1
010001 => [2,4,1] => [[5,5,2],[4,1]]
 => ? = 5
Description
The number of missing boxes of a skew partition.