- FindStat

edit this statistic or download as text // json

Identifier

St001909: Posets ⟶ ℤ

Values

([],1) => 2

([],2) => 4

([(0,1)],2) => 4

([],3) => 8

([(1,2)],3) => 8

([(0,1),(0,2)],3) => 8

([(0,2),(2,1)],3) => 7

([(0,2),(1,2)],3) => 8

([],4) => 16

([(2,3)],4) => 16

([(1,2),(1,3)],4) => 16

([(0,1),(0,2),(0,3)],4) => 16

([(0,2),(0,3),(3,1)],4) => 14

([(0,1),(0,2),(1,3),(2,3)],4) => 13

([(1,2),(2,3)],4) => 14

([(0,3),(3,1),(3,2)],4) => 13

([(1,3),(2,3)],4) => 16

([(0,3),(1,3),(3,2)],4) => 13

([(0,3),(1,3),(2,3)],4) => 16

([(0,3),(1,2)],4) => 16

([(0,3),(1,2),(1,3)],4) => 16

([(0,2),(0,3),(1,2),(1,3)],4) => 16

([(0,3),(2,1),(3,2)],4) => 11

([(0,3),(1,2),(2,3)],4) => 14

([],5) => 32

([(3,4)],5) => 32

([(2,3),(2,4)],5) => 32

([(1,2),(1,3),(1,4)],5) => 32

([(0,1),(0,2),(0,3),(0,4)],5) => 32

([(0,2),(0,3),(0,4),(4,1)],5) => 28

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 26

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 25

([(1,3),(1,4),(4,2)],5) => 28

([(0,3),(0,4),(4,1),(4,2)],5) => 26

([(1,2),(1,3),(2,4),(3,4)],5) => 26

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 19

([(0,3),(0,4),(3,2),(4,1)],5) => 25

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 24

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 23

([(2,3),(3,4)],5) => 28

([(1,4),(4,2),(4,3)],5) => 26

([(0,4),(4,1),(4,2),(4,3)],5) => 25

([(2,4),(3,4)],5) => 32

([(1,4),(2,4),(4,3)],5) => 26

([(0,4),(1,4),(4,2),(4,3)],5) => 23

([(1,4),(2,4),(3,4)],5) => 32

([(0,4),(1,4),(2,4),(4,3)],5) => 25

([(0,4),(1,4),(2,4),(3,4)],5) => 32

([(0,4),(1,4),(2,3)],5) => 32

([(0,4),(1,3),(2,3),(2,4)],5) => 32

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 32

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 32

([(0,4),(1,4),(2,3),(4,2)],5) => 19

([(0,4),(1,3),(2,3),(3,4)],5) => 26

([(0,4),(1,4),(2,3),(2,4)],5) => 32

([(0,4),(1,4),(2,3),(3,4)],5) => 28

([(1,4),(2,3)],5) => 32

([(1,4),(2,3),(2,4)],5) => 32

([(0,4),(1,2),(1,4),(2,3)],5) => 28

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 24

([(1,3),(1,4),(2,3),(2,4)],5) => 32

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 26

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 23

([(0,4),(1,2),(1,4),(4,3)],5) => 26

([(0,4),(1,2),(1,3)],5) => 32

([(0,4),(1,2),(1,3),(1,4)],5) => 32

([(0,2),(0,4),(3,1),(4,3)],5) => 22

([(0,4),(1,2),(1,3),(3,4)],5) => 28

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 21

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 26

([(0,3),(0,4),(1,2),(1,4)],5) => 32

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 32

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 32

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 28

([(0,3),(1,2),(1,4),(3,4)],5) => 28

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 26

([(1,4),(3,2),(4,3)],5) => 22

([(0,3),(3,4),(4,1),(4,2)],5) => 19

([(1,4),(2,3),(3,4)],5) => 28

([(0,4),(1,2),(2,4),(4,3)],5) => 21

([(0,3),(1,4),(4,2)],5) => 28

([(0,4),(3,2),(4,1),(4,3)],5) => 21

([(0,4),(1,2),(2,3),(2,4)],5) => 26

([(0,4),(2,3),(3,1),(4,2)],5) => 16

([(0,3),(1,2),(2,4),(3,4)],5) => 25

([(0,4),(1,2),(2,3),(3,4)],5) => 22

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 19

([],6) => 64

([(4,5)],6) => 64

([(3,4),(3,5)],6) => 64

([(2,3),(2,4),(2,5)],6) => 64

([(1,2),(1,3),(1,4),(1,5)],6) => 64

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 64

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 56

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 52

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => 50

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 49

([(1,3),(1,4),(1,5),(5,2)],6) => 56

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 52

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => 52

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 50

>>> Load all 405 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of interval-closed sets of a poset.
For a poset $P$ and a subset $I$ of $P$, we say that $I$ is an interval-closed set if for all $x, y \in I$ such that $x \leq y$, then $z \in I$ if $x \leq z \leq y$.
There is a bijection between interval-closed sets of a poset $P$ and pairs of disjoint antichains $(A, B)$ of $P$ such that any element in $B$ is in the order ideal $\Delta(A)$ generated by $A$. (Proposition 2.5 of [1]).

References

[1] Elder, J., Lafrenière, N., McNicholas, E., Striker, J., Welch, A. Toggling, rowmotion, and homomesy on interval-closed sets arXiv:2307.08520

Code

def statistic(x):
    ICS_count = 0
    for A in x.antichains_iterator():
        I = x.order_ideal(A)
        Q = x.subposet(set(I).difference(A))
        ICS_count += Q.antichains().cardinality()
    return ICS_count

Created

Jul 28, 2023 at 18:44 by Nadia Lafreniere

Updated

Jan 19, 2024 at 18:31 by Nadia Lafreniere