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Your data matches 151 different statistics following compositions of up to 3 maps.
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Matching statistic: St000058
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 2 = 1 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 3 = 2 + 1
[3,1,2] => 3 = 2 + 1
[3,2,1] => 2 = 1 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => 4 = 3 + 1
[2,4,1,3] => 4 = 3 + 1
[2,4,3,1] => 3 = 2 + 1
[3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => 4 = 3 + 1
[3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => 3 = 2 + 1
[3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => 4 = 3 + 1
[4,1,2,3] => 4 = 3 + 1
[4,1,3,2] => 3 = 2 + 1
[4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 4 = 3 + 1
[4,3,2,1] => 2 = 1 + 1
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000485
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(load all 16 compositions to match this statistic)
St000485: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 2 = 1 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 3 = 2 + 1
[3,1,2] => 3 = 2 + 1
[3,2,1] => 2 = 1 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => 4 = 3 + 1
[2,4,1,3] => 4 = 3 + 1
[2,4,3,1] => 3 = 2 + 1
[3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => 4 = 3 + 1
[3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => 3 = 2 + 1
[3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => 4 = 3 + 1
[4,1,2,3] => 4 = 3 + 1
[4,1,3,2] => 3 = 2 + 1
[4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 4 = 3 + 1
[4,3,2,1] => 2 = 1 + 1
Description
The length of the longest cycle of a permutation.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 2
[3,1,2] => [3]
=> 2
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [3,1]
=> 2
[1,4,2,3] => [3,1]
=> 2
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 1
[2,3,1,4] => [3,1]
=> 2
[2,3,4,1] => [4]
=> 3
[2,4,1,3] => [4]
=> 3
[2,4,3,1] => [3,1]
=> 2
[3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [4]
=> 3
[3,2,1,4] => [2,1,1]
=> 1
[3,2,4,1] => [3,1]
=> 2
[3,4,1,2] => [2,2]
=> 1
[3,4,2,1] => [4]
=> 3
[4,1,2,3] => [4]
=> 3
[4,1,3,2] => [3,1]
=> 2
[4,2,1,3] => [3,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 3
[4,3,2,1] => [2,2]
=> 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
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(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 2
[3,1,2] => [3]
=> 2
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [3,1]
=> 2
[1,4,2,3] => [3,1]
=> 2
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 1
[2,3,1,4] => [3,1]
=> 2
[2,3,4,1] => [4]
=> 3
[2,4,1,3] => [4]
=> 3
[2,4,3,1] => [3,1]
=> 2
[3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [4]
=> 3
[3,2,1,4] => [2,1,1]
=> 1
[3,2,4,1] => [3,1]
=> 2
[3,4,1,2] => [2,2]
=> 1
[3,4,2,1] => [4]
=> 3
[4,1,2,3] => [4]
=> 3
[4,1,3,2] => [3,1]
=> 2
[4,2,1,3] => [3,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 3
[4,3,2,1] => [2,2]
=> 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
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(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 2
[3,1,2] => [3]
=> 2
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [3,1]
=> 2
[1,4,2,3] => [3,1]
=> 2
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 1
[2,3,1,4] => [3,1]
=> 2
[2,3,4,1] => [4]
=> 3
[2,4,1,3] => [4]
=> 3
[2,4,3,1] => [3,1]
=> 2
[3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [4]
=> 3
[3,2,1,4] => [2,1,1]
=> 1
[3,2,4,1] => [3,1]
=> 2
[3,4,1,2] => [2,2]
=> 1
[3,4,2,1] => [4]
=> 3
[4,1,2,3] => [4]
=> 3
[4,1,3,2] => [3,1]
=> 2
[4,2,1,3] => [3,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 3
[4,3,2,1] => [2,2]
=> 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000147
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 1 = 0 + 1
[2,1] => [2]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [3]
=> 3 = 2 + 1
[3,1,2] => [3]
=> 3 = 2 + 1
[3,2,1] => [2,1]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> 3 = 2 + 1
[1,4,2,3] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => [3,1]
=> 3 = 2 + 1
[2,3,4,1] => [4]
=> 4 = 3 + 1
[2,4,1,3] => [4]
=> 4 = 3 + 1
[2,4,3,1] => [3,1]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> 3 = 2 + 1
[3,1,4,2] => [4]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> 2 = 1 + 1
[3,4,2,1] => [4]
=> 4 = 3 + 1
[4,1,2,3] => [4]
=> 4 = 3 + 1
[4,1,3,2] => [3,1]
=> 3 = 2 + 1
[4,2,1,3] => [3,1]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> 4 = 3 + 1
[4,3,2,1] => [2,2]
=> 2 = 1 + 1
Description
The largest part of an integer partition.
Matching statistic: St000668
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(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 1 = 0 + 1
[2,1] => [2]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [3]
=> 3 = 2 + 1
[3,1,2] => [3]
=> 3 = 2 + 1
[3,2,1] => [2,1]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> 3 = 2 + 1
[1,4,2,3] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => [3,1]
=> 3 = 2 + 1
[2,3,4,1] => [4]
=> 4 = 3 + 1
[2,4,1,3] => [4]
=> 4 = 3 + 1
[2,4,3,1] => [3,1]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> 3 = 2 + 1
[3,1,4,2] => [4]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> 2 = 1 + 1
[3,4,2,1] => [4]
=> 4 = 3 + 1
[4,1,2,3] => [4]
=> 4 = 3 + 1
[4,1,3,2] => [3,1]
=> 3 = 2 + 1
[4,2,1,3] => [3,1]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> 4 = 3 + 1
[4,3,2,1] => [2,2]
=> 2 = 1 + 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St001062
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> 1 = 0 + 1
[2,1] => {{1,2}}
=> 2 = 1 + 1
[1,2,3] => {{1},{2},{3}}
=> 1 = 0 + 1
[1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[2,3,1] => {{1,2,3}}
=> 3 = 2 + 1
[3,1,2] => {{1,2,3}}
=> 3 = 2 + 1
[3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> 2 = 1 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 2 = 1 + 1
[1,3,4,2] => {{1},{2,3,4}}
=> 3 = 2 + 1
[1,4,2,3] => {{1},{2,3,4}}
=> 3 = 2 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 1 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 2 = 1 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[2,3,1,4] => {{1,2,3},{4}}
=> 3 = 2 + 1
[2,3,4,1] => {{1,2,3,4}}
=> 4 = 3 + 1
[2,4,1,3] => {{1,2,3,4}}
=> 4 = 3 + 1
[2,4,3,1] => {{1,2,4},{3}}
=> 3 = 2 + 1
[3,1,2,4] => {{1,2,3},{4}}
=> 3 = 2 + 1
[3,1,4,2] => {{1,2,3,4}}
=> 4 = 3 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 1 + 1
[3,2,4,1] => {{1,3,4},{2}}
=> 3 = 2 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2 = 1 + 1
[3,4,2,1] => {{1,2,3,4}}
=> 4 = 3 + 1
[4,1,2,3] => {{1,2,3,4}}
=> 4 = 3 + 1
[4,1,3,2] => {{1,2,4},{3}}
=> 3 = 2 + 1
[4,2,1,3] => {{1,3,4},{2}}
=> 3 = 2 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 2 = 1 + 1
[4,3,1,2] => {{1,2,3,4}}
=> 4 = 3 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
Description
The maximal size of a block of a set partition.
Matching statistic: St001555
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001555: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [2,3,1] => 3 = 2 + 1
[3,1,2] => [3,1,2] => 3 = 2 + 1
[3,2,1] => [3,2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => [2,3,4,1] => 4 = 3 + 1
[2,4,1,3] => [2,4,1,3] => 4 = 3 + 1
[2,4,3,1] => [2,4,3,1] => 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => [3,1,4,2] => 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,4,2,1] => 4 = 3 + 1
[4,1,2,3] => [4,1,2,3] => 4 = 3 + 1
[4,1,3,2] => [4,1,3,2] => 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
Description
The order of a signed permutation.
Matching statistic: St000141
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 2
[3,1,2] => [3,2,1] => [3,2,1] => 2
[3,2,1] => [2,3,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 3
[2,4,3,1] => [3,4,1,2] => [2,4,1,3] => 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => 3
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,1,2] => [3,1,4,2] => [2,1,4,3] => 1
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 3
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,3,2] => [3,4,2,1] => [1,4,3,2] => 2
[4,2,1,3] => [2,4,3,1] => [1,4,3,2] => 2
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => 1
[4,3,1,2] => [4,2,3,1] => [4,1,3,2] => 3
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000956The maximal displacement of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000010The length of the partition. St000381The largest part of an integer composition. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000503The maximal difference between two elements in a common block. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001727The number of invisible inversions of a permutation. St000005The bounce statistic of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000062The length of the longest increasing subsequence of the permutation. St000288The number of ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000653The last descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000746The number of pairs with odd minimum in a perfect matching. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. 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. St001415The length of the longest palindromic prefix of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000054The first entry of the permutation. St000144The pyramid weight of the Dyck path. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000439The position of the first down step of a Dyck path. St000839The largest opener of a set partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001497The position of the largest weak excedence of a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000454The largest eigenvalue of a graph if it is integral. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000259The diameter of a connected graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000997The even-odd crank of an integer partition. St001571The Cartan determinant of the integer partition. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001118The acyclic chromatic index of a graph. St001624The breadth of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001330The hat guessing number of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001498The normalised height of a Nakayama algebra with magnitude 1. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001645The pebbling number of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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