Identifier
Values
01 => 00 => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
10 => 11 => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
010 => 000 => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
101 => 111 => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => ([(0,2),(2,1)],3) => 1
0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
1101 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => ([(0,2),(2,1)],3) => 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10) => ([(0,3),(2,1),(3,2)],4) => 1
00001 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => ([(0,2),(2,1)],3) => 1
00010 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
00100 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
01000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
01001 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
01101 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
01111 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => ([(0,2),(2,1)],3) => 1
10000 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => ([(0,2),(2,1)],3) => 1
10010 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
10110 => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
10111 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
11011 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
11101 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => ([(0,3),(2,1),(3,2)],4) => 1
11110 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => ([(0,2),(2,1)],3) => 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10) => ([(0,3),(2,1),(3,2)],4) => 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
000001 => 010100 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
000110 => 010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17) => ([(0,2),(2,1)],3) => 1
000111 => 010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15) => ([(0,2),(2,1)],3) => 1
001001 => 011100 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
001101 => 011000 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
001110 => 011011 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15) => ([(0,2),(2,1)],3) => 1
010011 => 000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
010101 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
011000 => 001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17) => ([(0,2),(2,1)],3) => 1
011011 => 001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
011100 => 001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15) => ([(0,2),(2,1)],3) => 1
011110 => 001011 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
011111 => 001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
100000 => 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
100001 => 110100 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
100011 => 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15) => ([(0,2),(2,1)],3) => 1
100100 => 110001 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
100111 => 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17) => ([(0,2),(2,1)],3) => 1
101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
101100 => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
110001 => 100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15) => ([(0,2),(2,1)],3) => 1
110010 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
110110 => 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
111000 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15) => ([(0,2),(2,1)],3) => 1
111001 => 101100 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17) => ([(0,2),(2,1)],3) => 1
111110 => 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
search for individual values
searching the database for the individual values of this statistic
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.
Map
antichains of maximal size
Description
The lattice of antichains of maximal size in a poset.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.