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Your data matches 245 different statistics following compositions of up to 3 maps.
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Matching statistic: St000183
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Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000473
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 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]].
Matching statistic: St001280
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001621
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001878
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001175
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001657
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 0 = 1 - 1
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001845
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0 = 1 - 1
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001906
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,3,5,4] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,2,5,3,4] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,4,5,3] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,5,3,4] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,3,5,4] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,2,4,3,5] => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,2,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,2,4] => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,2,3,5] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,5,2,3,4] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,4,5,2,3] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,5,3] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,5,2,3,4] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,5,2,3,4] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,5,2,4] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,5,2,4,3] => 0 = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0 = 1 - 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,2,5,3] => 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,3,4,2,5] => 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [1,4,2,3,5] => 0 = 1 - 1
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
The following 235 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000527The width of the poset. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001393The induced matching number of a graph. St001597The Frobenius rank of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000017The number of inversions of a standard tableau. St000057The Shynar inversion number of a standard tableau. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000142The number of even parts of a partition. St000317The cycle descent number of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000480The number of lower covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000807The sum of the heights of the valleys of the associated bargraph. St000931The number of occurrences of the pattern UUU in a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001261The Castelnuovo-Mumford regularity of a graph. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001596The number of two-by-two squares inside a skew partition. St001728The number of invisible descents of a permutation. St001110The 3-dynamic chromatic number of a graph. St001432The order dimension of the partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St000010The length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000147The largest part of an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000346The number of coarsenings of a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000481The number of upper covers of a partition in dominance order. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in 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. 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. St000632The jump number of the 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001271The competition number of a graph. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001128The exponens consonantiae of a partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001715The number of non-records in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000256The number of parts from which one can substract 2 and still get an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000668The least common multiple of the parts of the partition. St000741The Colin de Verdière graph invariant. St001890The maximum magnitude of the Möbius function of a poset. St001644The dimension of a graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001256Number of simple reflexive modules that are 2-stable reflexive. St001638The book thickness of a graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001545The second Elser number of a connected graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000655The length of the minimal rise of a Dyck path. St000297The number of leading ones in a binary word. St001625The Möbius invariant of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001520The number of strict 3-descents. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001344The neighbouring number of a permutation. St001645The pebbling number of a connected graph. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001734The lettericity of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000879The number of long braid edges in the graph of braid moves of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000768The number of peaks in an integer composition. St001429The number of negative entries in a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000068The number of minimal elements in a poset. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000291The number of descents of a binary word. St000629The defect of a binary word. St000390The number of runs of ones in a binary word. St001725The harmonious chromatic number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001330The hat guessing number of a graph. St000382The first part of an integer composition. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001828The Euler characteristic of a graph. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000822The Hadwiger number of the graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St001117The game chromatic index of a graph. St000286The number of connected components of the complement of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000379The number of Hamiltonian cycles in a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001339The irredundance number of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001866The nesting alignments of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word.
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