Your data matches 49 different statistics following compositions of up to 3 maps.
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Mp00013: Binary trees to posetPosets
St000307: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
Description
The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000010
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 57% values known / values provided: 90%distinct values known / distinct values provided: 57%
Values
[.,.]
=> ([],1)
=> [2]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> ? = 5
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> ? = 5
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[.,[.,.]],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> [6,6,4,3,3,2,2]
=> ? = 7
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[[[.,.],.],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,[.,.]],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[[.,.],.],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[.,[[.,.],[.,.]]],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],[[.,.],.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,[.,.]],[.,.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[[.,.],.],[.,.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[[.,.],[.,.]],.],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[[.,.],[.,.]],.],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[[.,.],[.,[[.,.],.]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[[.,.],[[.,.],[.,.]]],[.,.]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> ? = 5
Description
The length of the partition.
Matching statistic: St001280
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St001280: Integer partitions ⟶ ℤResult quality: 57% values known / values provided: 90%distinct values known / distinct values provided: 57%
Values
[.,.]
=> ([],1)
=> [2]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> ? = 5
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> ? = 4
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> ? = 5
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[.,[.,.]],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> [6,6,4,3,3,2,2]
=> ? = 7
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,.],[.,.]],[[[.,.],.],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[.,[.,.]],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[[[.,.],.],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> ? = 3
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[.,[[.,.],[.,.]]],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,.],[[.,.],.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[.,[.,.]],[.,.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[[.,.],.],[.,.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> ? = 1
[[[[.,.],[.,.]],.],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[[.,.],[.,.]],.],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> ? = 2
[[[.,.],[.,[.,[.,.]]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[[.,.],[.,[[.,.],.]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> ? = 3
[[[.,.],[[.,.],[.,.]]],[.,.]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> ? = 5
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001918
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St001918: Integer partitions ⟶ ℤResult quality: 57% values known / values provided: 88%distinct values known / distinct values provided: 57%
Values
[.,.]
=> ([],1)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> [5,5,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 5 - 1
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[.,.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4 - 1
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4 - 1
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4 - 1
[[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4 - 1
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> [5,5,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 5 - 1
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[[[.,.],[[.,.],.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,.],[[[[.,.],.],[.,.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2 - 1
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[[.,[.,.]],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[.,[.,.]],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[.,[.,.]],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> [6,6,4,3,3,2,2]
=> [7,7,5,3,2,2]
=> ? = 7 - 1
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[[.,.],[.,.]],[[[.,.],.],.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[[.,[.,.]],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[[[.,.],.],.],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[[.,[[.,.],[.,.]]],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],[[.,.],.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,.],[[.,.],.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
[[[.,[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000147
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 57% values known / values provided: 84%distinct values known / distinct values provided: 57%
Values
[.,.]
=> ([],1)
=> [2]
=> [1,1]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 4
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 4
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 4
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 4
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> [5,5,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 5
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[.,.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4
[[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> [6,6,5,4]
=> [4,4,4,4,3,2]
=> ? = 4
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [15,2,2,2,2]
=> [5,5,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 5
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> [11,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1]
=> ? = 3
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[[[.,.],[[.,.],.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,.],[[[[.,.],.],[.,.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 2
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[[.,[.,.]],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> [22]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> [13,6]
=> [2,2,2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2
[[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[.,[.,.]]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3
[[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[.,[.,.]]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[[.,.],.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[[.,.],.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3
[[.,[[.,.],.]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[.,[[.,.],.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> ? = 4
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> [13,4,4]
=> [3,3,3,3,1,1,1,1,1,1,1,1,1]
=> ? = 3
Description
The largest part of an integer partition.
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St000533: Integer partitions ⟶ ℤResult quality: 57% values known / values provided: 58%distinct values known / distinct values provided: 57%
Values
[.,.]
=> ([],1)
=> [2]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
Description
The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000278
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St000278: Integer partitions ⟶ ℤResult quality: 43% values known / values provided: 57%distinct values known / distinct values provided: 43%
Values
[.,.]
=> ([],1)
=> [2]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$, where $k$ is the number of parts of $\lambda$. An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
Matching statistic: St000473
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St000473: Integer partitions ⟶ ℤResult quality: 43% values known / values provided: 57%distinct values known / distinct values provided: 43%
Values
[.,.]
=> ([],1)
=> [2]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
Description
The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
St001432: Integer partitions ⟶ ℤResult quality: 43% values known / values provided: 57%distinct values known / distinct values provided: 43%
Values
[.,.]
=> ([],1)
=> [2]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 2
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 4
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 4
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> ? = 3
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> ? = 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> ? = 4
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> ? = 1
Description
The order dimension of the partition. Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000668
Mp00013: Binary trees to posetPosets
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000668: Integer partitions ⟶ ℤResult quality: 43% values known / values provided: 57%distinct values known / distinct values provided: 43%
Values
[.,.]
=> ([],1)
=> [2]
=> [1,1]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 2
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 4
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 4
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 4
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 4
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 4
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> ? = 4
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 4
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 4
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> ? = 3
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> ? = 3
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 4
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 4
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [6,5,3,3]
=> [4,4,4,2,2,1]
=> ? = 4
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
Description
The least common multiple of the parts of the partition.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001128The exponens consonantiae of a partition. St000146The Andrews-Garvan crank of a partition. St000783The side length of the largest staircase partition fitting into a partition. St000143The largest repeated part of a partition. St000225Difference between largest and smallest parts in a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000100The number of linear extensions of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset.