Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001437
Mapping
Mp00278: Binary words —rowmotion⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1
1 => 1 => 1
00 => 00 => 2
01 => 10 => 2
10 => 01 => 1
11 => 11 => 2
000 => 000 => 3
001 => 010 => 2
010 => 100 => 3
011 => 101 => 2
100 => 001 => 1
101 => 110 => 3
110 => 011 => 1
111 => 111 => 3
0000 => 0000 => 4
0001 => 0010 => 2
0010 => 0100 => 3
0011 => 0101 => 2
0100 => 1000 => 4
0101 => 1010 => 4
0110 => 1001 => 3
0111 => 1011 => 2
1000 => 0001 => 1
1001 => 0110 => 2
1010 => 1100 => 4
1011 => 1101 => 3
1100 => 0011 => 1
1101 => 1110 => 4
1110 => 0111 => 1
1111 => 1111 => 4
00000 => 00000 => 5
00001 => 00010 => 2
00010 => 00100 => 3
00011 => 00101 => 1
00100 => 01000 => 4
00101 => 01010 => 3
00110 => 01001 => 2
00111 => 01011 => 1
01000 => 10000 => 5
01001 => 10010 => 4
01010 => 10100 => 5
01011 => 10101 => 3
01100 => 10001 => 4
01101 => 10110 => 4
01110 => 10011 => 3
01111 => 10111 => 2
10000 => 00001 => 1
10001 => 00110 => 2
10010 => 01100 => 3
10011 => 01101 => 2
Description
The flex of a binary word. This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St001880
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 38%
Values
0 => [1] => [[1],[]]
 => ([],1)
 => ? = 1
1 => [1] => [[1],[]]
 => ([],1)
 => ? = 1
00 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? = 2
01 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 2
10 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 1
11 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? = 2
000 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
001 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 2
010 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
011 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 2
100 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 1
101 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 3
110 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 1
111 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 3
0000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
0001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
0011 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 2
0100 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 4
0101 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
0110 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 3
0111 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 2
1000 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 1
1001 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 2
1010 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
1011 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
1100 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4
1110 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
1111 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
00000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
00001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3
00011 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
00111 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 1
01000 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 5
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
01011 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 3
01100 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? = 4
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 4
01110 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 3
01111 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 2
10000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 1
10001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 2
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 3
10011 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? = 2
10100 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
10111 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 3
11000 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 1
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 5
11011 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4
11100 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 1
11101 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 5
11110 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1
11111 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
000000 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
101010 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
111111 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
0000000 => [7] => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
0101010 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
1010101 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
1111111 => [7] => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 38%
Values
0 => [1] => [[1],[]]
 => ([],1)
 => ? = 1 - 1
1 => [1] => [[1],[]]
 => ([],1)
 => ? = 1 - 1
00 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? = 2 - 1
01 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 2 - 1
10 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => ? = 1 - 1
11 => [2] => [[2],[]]
 => ([(0,1)],2)
 => ? = 2 - 1
000 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
001 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 2 - 1
010 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
011 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 2 - 1
100 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ? = 1 - 1
101 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
110 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => ? = 1 - 1
111 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
0000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
0001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 1
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
0011 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 2 - 1
0100 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 4 - 1
0101 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
0110 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
0111 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 2 - 1
1000 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 1 - 1
1001 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 2 - 1
1010 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
1011 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ? = 3 - 1
1100 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 1 - 1
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 4 - 1
1110 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 1
1111 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
00000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
00001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 - 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 - 1
00011 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 1 - 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4 - 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 1
00111 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 1 - 1
01000 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 5 - 1
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 1
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
01011 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 3 - 1
01100 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? = 4 - 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 4 - 1
01110 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 3 - 1
01111 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 2 - 1
10000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 1 - 1
10001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 2 - 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 3 - 1
10011 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ? = 2 - 1
10100 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 5 - 1
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4 - 1
10111 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 3 - 1
11000 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 1 - 1
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 1
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 5 - 1
11011 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? = 4 - 1
11100 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 1 - 1
11101 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 5 - 1
11110 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 1 - 1
11111 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
000000 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
101010 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
111111 => [6] => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
0000000 => [7] => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
0101010 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
1010101 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
1111111 => [7] => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.