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Your data matches 469 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000706
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001933
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000150
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 0 = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000212
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000212: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000212: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 0 = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
Description
The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row.
Summing over all partitions of $n$ yields the sequence
$$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$
which is [[oeis:A237770]].
The references in this sequence of the OEIS indicate a connection with Baxter permutations.
Matching statistic: St000257
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 0 = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St001091
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 0 = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 1 - 1
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000667
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000149
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
The following 459 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000478Another weight of a partition according to Alladi. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001571The Cartan determinant of the integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001890The maximum magnitude of the Möbius function of a poset. St001568The smallest positive integer that does not appear twice in the partition. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. 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. St001618The cardinality of the Frattini sublattice of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001260The permanent of an alternating sign matrix. St001116The game chromatic number of a graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000546The number of global descents of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001638The book thickness of a graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000068The number of minimal elements in a poset. St000623The number of occurrences of the pattern 52341 in a permutation. St001381The fertility of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000665The number of rafts of a permutation. St000862The number of parts of the shifted shape of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000363The number of minimal vertex covers of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001342The number of vertices in the center of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000098The chromatic number of a graph. St000268The number of strongly connected orientations of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001316The domatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001108The 2-dynamic chromatic number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000916The packing number of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000124The cardinality of the preimage of the Simion-Schmidt map. St000258The burning number of a graph. St000259The diameter of a connected graph. St000273The domination number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001829The common independence number of a graph. St000264The girth of a graph, which is not a tree. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000666The number of right tethers of a permutation. St001130The number of two successive successions in a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. 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 $K[x]/(x^n)$. St001811The Castelnuovo-Mumford regularity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000056The decomposition (or block) number of a permutation. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000694The number of affine bounded permutations that project to a given permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001765The number of connected components of the friends and strangers graph. St001827The number of two-component spanning forests of a graph. St001828The Euler characteristic of a graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000095The number of triangles of a graph. St000221The number of strong fixed points of a permutation. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000822The Hadwiger number of the graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001060The distinguishing index of a graph. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001429The number of negative entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. 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. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001871The number of triconnected components of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000061The number of nodes on the left branch of a binary tree. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000326The position of the first one in a binary word after appending a 1 at the end. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000553The number of blocks of a graph. St000570The Edelman-Greene number of a permutation. St000627The exponent of a binary word. St000654The first descent of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000843The decomposition number of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000917The open packing number of a graph. St000958The number of Bruhat factorizations of a permutation. St000991The number of right-to-left minima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001344The neighbouring number of a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001393The induced matching number of a graph. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001739The number of graphs with the same edge polytope as the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000210Minimum over maximum difference of elements in cycles. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000308The height of the tree associated to a permutation. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000461The rix statistic of a permutation. St000477The weight of a partition according to Alladi. St000481The number of upper covers of a partition in dominance order. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000629The defect of a binary word. St000664The number of right ropes of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000732The number of double deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000750The number of occurrences of the pattern 4213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001261The Castelnuovo-Mumford regularity of a graph. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001307The number of induced stars on four vertices in a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001471The magnitude of a Dyck path. St001479The number of bridges of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001513The number of nested exceedences of a permutation. St001530The depth of a Dyck path. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001555The order of a signed permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001826The maximal number of leaves on a vertex of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000159The number of distinct parts of the integer partition. St001119The length of a shortest maximal path in a graph. St001315The dissociation number of a graph. St001432The order dimension of the partition. St001488The number of corners of a skew partition. St001642The Prague dimension of a graph. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001110The 3-dynamic chromatic number of a graph. 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. St001345The Hamming dimension of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension 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. 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. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001644The dimension of a graph. St000635The number of strictly order preserving maps of a poset into itself. St000741The Colin de Verdière graph invariant. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001625The Möbius invariant of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001783The number of odd automorphisms of a graph. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset.
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