Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001313
(load all 3 compositions to match this statistic)
Mapping
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 1
01 => 2
10 => 1
11 => 1
000 => 1
001 => 3
010 => 2
011 => 3
100 => 1
101 => 2
110 => 1
111 => 1
0000 => 1
0001 => 4
0010 => 3
0011 => 6
0100 => 2
0101 => 5
0110 => 3
0111 => 4
1000 => 1
1001 => 3
1010 => 2
1011 => 3
1100 => 1
1101 => 2
1110 => 1
1111 => 1
00000 => 1
00001 => 5
00010 => 4
00011 => 10
00100 => 3
00101 => 9
00110 => 6
00111 => 10
01000 => 2
01001 => 7
01010 => 5
01011 => 9
01100 => 3
01101 => 7
01110 => 4
01111 => 5
10000 => 1
10001 => 4
10010 => 3
10011 => 6
Description
The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$. See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St000108
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 31%
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]
 => 3
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 2
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 3
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]
 => 4
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 3
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 6
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 5
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 3
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 4
1000 => [1,4] => [[4,1],[]]
 => []
 => 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 3
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 3
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]
 => 5
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 4
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 10
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 3
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 9
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 6
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 10
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 7
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 5
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 9
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 3
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 7
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 4
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 5
10000 => [1,5] => [[5,1],[]]
 => []
 => 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 4
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 3
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 6
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [4,4,4]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => [4,4,3]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [3,3,3,3]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => [3,3,3,2]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1001000 => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1001111 => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1010000 => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1011000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1011100 => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1101000 => [1,1,2,4] => [[5,2,1,1],[1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1101100 => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1110001 => [1,1,1,4,1] => [[4,4,1,1,1],[3]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1110011 => [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1110100 => [1,1,1,2,3] => [[4,2,1,1,1],[1]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
1111001 => [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
 => ?
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
00000010 => [7,2] => [[8,7],[6]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00000011 => [7,1,1] => [[7,7,7],[6,6]]
 => [6,6]
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00000100 => [6,3] => [[8,6],[5]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00000101 => [6,2,1] => [[7,7,6],[6,5]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00000110 => [6,1,2] => [[7,6,6],[5,5]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00000111 => [6,1,1,1] => [[6,6,6,6],[5,5,5]]
 => [5,5,5]
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001000 => [5,4] => [[8,5],[4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001001 => [5,3,1] => [[7,7,5],[6,4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001010 => [5,2,2] => [[7,6,5],[5,4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001011 => [5,2,1,1] => [[6,6,6,5],[5,5,4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001100 => [5,1,3] => [[7,5,5],[4,4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001101 => [5,1,2,1] => [[6,6,5,5],[5,4,4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
00001110 => [5,1,1,2] => [[6,5,5,5],[4,4,4]]
 => ?
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
Description
The number of partitions contained in the given partition.