- FindStat

Identifier

Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001876: Lattices ⟶ ℤ (values match St001877Number of indecomposable injective modules with projective dimension 2.)

Values

=> 0 => ([(0,1)],2) => ([(0,2),(2,1)],3) => 0
=> 1 => ([(0,1)],2) => ([(0,2),(2,1)],3) => 0
=> 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
=> 00 => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 0
=> 11 => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 0
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9) => 2
=> 000 => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
=> 111 => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9) => 2
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
=> 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
010101 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 0
101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 0
0101010 => 0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 0
1010101 => 1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 0

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$F_{1} = 2$

$F_{2} = 2 + 2\ q$

Description

The number of 2-regular simple modules in the incidence algebra of the lattice.

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

$v$ .

Map

order ideals

Description

The lattice of order ideals of a poset.
An order ideal

$\mathcal I$ in a poset

$P$ is a downward closed set, i.e.,

$a \in \mathcal I$ and

$b \leq a$ implies

$b \in \mathcal I$ . This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.

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.