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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000480
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,2] => [1,1]
=> 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [3,1,2] => [1,2,3] => [1,1,1]
=> 0
[2,1,3] => [2,3,1] => [1,2,3] => [1,1,1]
=> 0
[2,3,1] => [2,1,3] => [1,3,2] => [2,1]
=> 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St001124
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,2] => [1,1]
=> 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [3,1,2] => [1,2,3] => [1,1,1]
=> 0
[2,1,3] => [2,3,1] => [1,2,3] => [1,1,1]
=> 0
[2,3,1] => [2,1,3] => [1,3,2] => [2,1]
=> 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
=> 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000159
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => [1,1]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[2,1,3] => [2,3,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[2,3,1] => [2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000783
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => [1,1]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[2,1,3] => [2,3,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[2,3,1] => [2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001432
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => [1,1]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[2,1,3] => [2,3,1] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[2,3,1] => [2,1,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
=> 2 = 1 + 1
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000318
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,2] => [1,1]
=> 2 = 0 + 2
[2,1] => [1,2] => [1,2] => [1,1]
=> 2 = 0 + 2
[1,2,3] => [3,2,1] => [1,2,3] => [1,1,1]
=> 2 = 0 + 2
[1,3,2] => [3,1,2] => [1,2,3] => [1,1,1]
=> 2 = 0 + 2
[2,1,3] => [2,3,1] => [1,2,3] => [1,1,1]
=> 2 = 0 + 2
[2,3,1] => [2,1,3] => [1,3,2] => [2,1]
=> 3 = 1 + 2
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 3 = 1 + 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 2 = 0 + 2
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 3 = 1 + 2
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 1 + 2
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 3 = 1 + 2
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 3 = 1 + 2
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 3 = 1 + 2
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 3 = 1 + 2
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 3 = 1 + 2
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 3 = 1 + 2
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 3 = 1 + 2
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 3 = 1 + 2
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 3 = 1 + 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3 = 1 + 2
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3 = 1 + 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 2 = 0 + 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 2 = 0 + 2
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => [2,1,1,1]
=> 3 = 1 + 2
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => [2,1,1,1]
=> 3 = 1 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000485
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Values
[1,2] => [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1,3] => [3,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2 = 1 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[2,3,4,5,1,6,7] => [7,6,1,5,4,3,2] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[2,3,4,5,1,7,6] => [6,7,1,5,4,3,2] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[2,3,4,5,6,1,7] => [7,1,6,5,4,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,4,5,7,1,6] => [6,1,7,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,4,5,7,6,1] => [1,6,7,5,4,3,2] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 1 + 1
[2,3,4,6,1,5,7] => [7,5,1,6,4,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,4,6,1,7,5] => [5,7,1,6,4,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,4,6,5,1,7] => [7,1,5,6,4,3,2] => [1,5,6,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[2,3,4,6,5,7,1] => [1,7,5,6,4,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,4,6,7,1,5] => [5,1,7,6,4,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,4,6,7,5,1] => [1,5,7,6,4,3,2] => [1,5,7,2,3,4,6] => [1,6,7,4,5,2,3] => ? = 1 + 1
[2,3,4,7,1,5,6] => [6,5,1,7,4,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,4,7,1,6,5] => [5,6,1,7,4,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,4,7,5,1,6] => [6,1,5,7,4,3,2] => [1,5,7,2,3,4,6] => [1,6,7,4,5,2,3] => ? = 1 + 1
[2,3,4,7,5,6,1] => [1,6,5,7,4,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,4,7,6,1,5] => [5,1,6,7,4,3,2] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 1 + 1
[2,3,4,7,6,5,1] => [1,5,6,7,4,3,2] => [1,5,6,7,2,3,4] => [1,7,6,5,4,3,2] => ? = 1 + 1
[2,3,5,1,4,6,7] => [7,6,4,1,5,3,2] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[2,3,5,1,4,7,6] => [6,7,4,1,5,3,2] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[2,3,5,1,6,4,7] => [7,4,6,1,5,3,2] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[2,3,5,1,6,7,4] => [4,7,6,1,5,3,2] => [1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => ? = 1 + 1
[2,3,5,1,7,4,6] => [6,4,7,1,5,3,2] => [1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => ? = 1 + 1
[2,3,5,1,7,6,4] => [4,6,7,1,5,3,2] => [1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => ? = 1 + 1
[2,3,5,4,1,6,7] => [7,6,1,4,5,3,2] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[2,3,5,4,1,7,6] => [6,7,1,4,5,3,2] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[2,3,5,4,6,1,7] => [7,1,6,4,5,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,5,4,6,7,1] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,4,7,1,6] => [6,1,7,4,5,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,4,7,6,1] => [1,6,7,4,5,3,2] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 1 + 1
[2,3,5,6,1,4,7] => [7,4,1,6,5,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,5,6,1,7,4] => [4,7,1,6,5,3,2] => [1,6,2,3,4,7,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,6,4,1,7] => [7,1,4,6,5,3,2] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[2,3,5,6,4,7,1] => [1,7,4,6,5,3,2] => [1,7,2,3,4,6,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,6,7,1,4] => [4,1,7,6,5,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,6,7,4,1] => [1,4,7,6,5,3,2] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 1 + 1
[2,3,5,7,1,4,6] => [6,4,1,7,5,3,2] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,7,1,6,4] => [4,6,1,7,5,3,2] => [1,7,2,3,4,6,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,7,4,1,6] => [6,1,4,7,5,3,2] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 1 + 1
[2,3,5,7,4,6,1] => [1,6,4,7,5,3,2] => [1,6,2,3,4,7,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,5,7,6,1,4] => [4,1,6,7,5,3,2] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 1 + 1
[2,3,5,7,6,4,1] => [1,4,6,7,5,3,2] => [1,4,6,7,2,3,5] => [1,6,7,5,4,2,3] => ? = 1 + 1
[2,3,6,1,4,5,7] => [7,5,4,1,6,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,6,1,4,7,5] => [5,7,4,1,6,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,6,1,5,4,7] => [7,4,5,1,6,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,6,1,5,7,4] => [4,7,5,1,6,3,2] => [1,6,2,3,4,7,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,6,1,7,4,5] => [5,4,7,1,6,3,2] => [1,6,2,3,4,7,5] => [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,6,1,7,5,4] => [4,5,7,1,6,3,2] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 1 + 1
[2,3,6,4,1,5,7] => [7,5,1,4,6,3,2] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
[2,3,6,4,1,7,5] => [5,7,1,4,6,3,2] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 1 + 1
Description
The length of the longest cycle of a permutation.
Matching statistic: St001174
(load all 115 compositions to match this statistic)
(load all 115 compositions to match this statistic)
St001174: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
[1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => ? = 0
[1,2,3,4,6,5,7] => ? = 0
[1,2,3,4,6,7,5] => ? = 1
[1,2,3,4,7,5,6] => ? = 1
[1,2,3,4,7,6,5] => ? = 0
[1,2,3,5,4,6,7] => ? = 0
[1,2,3,5,4,7,6] => ? = 0
[1,2,3,5,6,4,7] => ? = 1
[1,2,3,5,6,7,4] => ? = 1
[1,2,3,5,7,4,6] => ? = 1
[1,2,3,5,7,6,4] => ? = 1
[1,2,3,6,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => ? = 1
[1,2,3,6,5,4,7] => ? = 0
[1,2,3,6,5,7,4] => ? = 1
[1,2,3,6,7,4,5] => ? = 1
[1,2,3,6,7,5,4] => ? = 1
[1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,4,6,5] => ? = 1
[1,2,3,7,5,4,6] => ? = 1
[1,2,3,7,5,6,4] => ? = 1
[1,2,3,7,6,4,5] => ? = 1
[1,2,3,7,6,5,4] => ? = 0
[1,2,4,3,5,6,7] => ? = 0
[1,2,4,3,5,7,6] => ? = 0
[1,2,4,3,6,5,7] => ? = 0
[1,2,4,3,6,7,5] => ? = 1
[1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,6,5] => ? = 0
[1,2,4,5,3,6,7] => ? = 1
[1,2,4,5,3,7,6] => ? = 1
[1,2,4,5,6,3,7] => ? = 1
[1,2,4,5,6,7,3] => ? = 1
[1,2,4,5,7,3,6] => ? = 1
[1,2,4,5,7,6,3] => ? = 1
[1,2,4,6,3,5,7] => ? = 1
[1,2,4,6,3,7,5] => ? = 1
[1,2,4,6,5,3,7] => ? = 1
[1,2,4,6,5,7,3] => ? = 1
[1,2,4,6,7,3,5] => ? = 1
[1,2,4,6,7,5,3] => ? = 1
[1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,3,6,5] => ? = 1
[1,2,4,7,5,3,6] => ? = 1
[1,2,4,7,5,6,3] => ? = 1
[1,2,4,7,6,3,5] => ? = 1
[1,2,4,7,6,5,3] => ? = 1
[1,2,5,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => ? = 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001859
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [3,1,2] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,4,1,2] => 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,4,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,4,1,2] => 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,4,1,2] => 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,4,1,2] => 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,4,1,2] => 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,1,2,4] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,5,1,2,3] => 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
Description
The number of factors of the Stanley symmetric function associated with a permutation.
For example, the Stanley symmetric function of $\pi=321645$ equals
$20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$
Matching statistic: St000308
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1,2] => [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[2,1,3] => [3,1,2] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,3,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => 2 = 1 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 + 1
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 + 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 + 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 + 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 + 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 1 + 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 + 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 + 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 + 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 + 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 1 + 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 + 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St000298The order dimension or Dushnik-Miller dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001060The distinguishing index of a graph. St000699The toughness times the least common multiple of 1,. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function.
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