searching the database
Your data matches 101 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => 2 = 0 + 2
[1,3,2] => 2 = 0 + 2
[2,1,3] => 2 = 0 + 2
[2,3,1] => 2 = 0 + 2
[3,1,2] => 2 = 0 + 2
[3,2,1] => 2 = 0 + 2
[1,2,3,4] => 3 = 1 + 2
[1,2,4,3] => 2 = 0 + 2
[1,3,2,4] => 3 = 1 + 2
[1,3,4,2] => 2 = 0 + 2
[1,4,2,3] => 3 = 1 + 2
[1,4,3,2] => 3 = 1 + 2
[2,1,3,4] => 2 = 0 + 2
[2,1,4,3] => 3 = 1 + 2
[2,3,1,4] => 3 = 1 + 2
[2,3,4,1] => 3 = 1 + 2
[2,4,1,3] => 3 = 1 + 2
[2,4,3,1] => 2 = 0 + 2
[3,1,2,4] => 2 = 0 + 2
[3,1,4,2] => 3 = 1 + 2
[3,2,1,4] => 3 = 1 + 2
[3,2,4,1] => 3 = 1 + 2
[3,4,1,2] => 3 = 1 + 2
[3,4,2,1] => 2 = 0 + 2
[4,1,2,3] => 3 = 1 + 2
[4,1,3,2] => 3 = 1 + 2
[4,2,1,3] => 2 = 0 + 2
[4,2,3,1] => 3 = 1 + 2
[4,3,1,2] => 2 = 0 + 2
[4,3,2,1] => 3 = 1 + 2
[1,2,3,4,5] => 4 = 2 + 2
[1,2,3,5,4] => 3 = 1 + 2
[1,2,4,3,5] => 4 = 2 + 2
[1,2,4,5,3] => 2 = 0 + 2
[1,2,5,3,4] => 3 = 1 + 2
[1,2,5,4,3] => 3 = 1 + 2
[1,3,2,4,5] => 4 = 2 + 2
[1,3,2,5,4] => 3 = 1 + 2
[1,3,4,2,5] => 4 = 2 + 2
[1,3,4,5,2] => 3 = 1 + 2
[1,3,5,2,4] => 3 = 1 + 2
[1,3,5,4,2] => 2 = 0 + 2
[1,4,2,3,5] => 4 = 2 + 2
[1,4,2,5,3] => 3 = 1 + 2
[1,4,3,2,5] => 4 = 2 + 2
[1,4,3,5,2] => 3 = 1 + 2
[1,4,5,2,3] => 3 = 1 + 2
[1,4,5,3,2] => 3 = 1 + 2
[1,5,2,3,4] => 4 = 2 + 2
[1,5,2,4,3] => 4 = 2 + 2
Description
The cyclic maximal difference between two consecutive entries of a permutation.
This is given, for a permutation $\pi$ of length $n$, by
$$\max \{ |\pi(i) − \pi(i+1)| : 1 \leq i \leq n \}$$
where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St001246
Mp00252: Permutations —restriction⟶ Permutations
St001246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => 1 = 0 + 1
[2,3,1] => [2,1] => 1 = 0 + 1
[3,1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,5,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,5,2,4,3] => [1,2,4,3] => 2 = 1 + 1
Description
The maximal difference between two consecutive entries of a permutation.
This is given, for a permutation $\pi$ of length $n$, by
$$\max\{ | \pi(i) - \pi(i+1) | : 1 \leq i < n \}.$$
Matching statistic: St000372
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [1,2] => 0
[2,3,1] => [2,1] => [1,2] => 0
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [1,2] => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,2,3] => 1
[1,3,4,2] => [1,3,2] => [1,2,3] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,2,3] => 1
[2,1,3,4] => [2,1,3] => [1,2,3] => 1
[2,1,4,3] => [2,1,3] => [1,2,3] => 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,2,3] => 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,3,2] => 0
[3,1,4,2] => [3,1,2] => [1,3,2] => 0
[3,2,1,4] => [3,2,1] => [1,3,2] => 0
[3,2,4,1] => [3,2,1] => [1,3,2] => 0
[3,4,1,2] => [3,1,2] => [1,3,2] => 0
[3,4,2,1] => [3,2,1] => [1,3,2] => 0
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,2,3] => 1
[4,2,1,3] => [2,1,3] => [1,2,3] => 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,3,2] => 0
[4,3,2,1] => [3,2,1] => [1,3,2] => 0
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 2
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 2
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2
[1,5,2,4,3] => [1,2,4,3] => [1,2,3,4] => 2
Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Matching statistic: St000371
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [3,2,1] => [2,1,3] => 0
[1,3,2] => [1,2,3] => [3,2,1] => [2,1,3] => 0
[2,1,3] => [1,2,3] => [3,2,1] => [2,1,3] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [3,2,1,4] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,2,1,3] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1,4] => 0
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1,4] => 0
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [2,3,1,4] => 0
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [2,3,1,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [4,3,1,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [4,3,1,2] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [4,3,1,2] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [4,3,1,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1,5] => 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [5,3,2,1,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,1,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [3,5,2,1,4] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,1,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [3,5,2,1,4] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,1,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,1,5] => 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [4,5,2,1,3] => 1
[1,5,2,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [5,4,2,1,3] => 2
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001683
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [2,1,3] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [2,4,1,3] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [2,3,4,1] => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [2,3,4,1] => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [2,1,3,4] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [2,1,4,3] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [2,1,3,4] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [2,1,4,3] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [2,1,4,3] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [2,1,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,4,3] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,4,3] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [2,5,4,1,3] => 2
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,4,3] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,4,3] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [2,5,1,3,4] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [2,5,3,4,1] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [2,5,1,3,4] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [2,5,3,4,1] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [2,5,1,3,4] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [2,5,1,3,4] => 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,3,4] => 1
[1,5,2,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,5,4,3] => 2
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St001960
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 2
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000316
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1,3] => [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[2,3,1] => [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[3,2,1] => [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,4,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[3,1,4,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[3,2,1,4] => [3,2,1] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[3,4,2,1] => [3,2,1] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[4,1,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[4,3,2,1] => [3,2,1] => [1,3,2] => [2,3,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [2]
=> 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
=> [2]
=> 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [2]
=> 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
=> [2]
=> 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 2
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [2]
=> 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
=> [2]
=> 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 2
Description
The product of the parts of an integer partition.
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000717The number of ordinal summands of a poset. St000080The rank of the poset. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000632The jump number of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001557The number of inversions of the second entry of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000681The Grundy value of Chomp on Ferrers diagrams. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001623The number of doubly irreducible elements of a lattice. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. 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. St000245The number of ascents of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001845The number of join irreducibles minus the rank of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000097The order of the largest clique of the graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001866The nesting alignments of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000778The metric dimension of a graph. St000482The (zero)-forcing number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000544The cop number of a graph. St001868The number of alignments of type NE of a signed permutation. St001935The number of ascents in a parking function. St000075The orbit size of a standard tableau under promotion. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001861The number of Bruhat lower covers of a permutation. St001863The number of weak excedances of a signed permutation. St001581The achromatic number of a graph. St001267The length of the Lyndon factorization of the binary word. St001621The number of atoms of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001875The number of simple modules with projective dimension at most 1. St000447The number of pairs of vertices of a graph with distance 3. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001734The lettericity of a graph. St000287The number of connected components of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001828The Euler characteristic of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001871The number of triconnected components of a graph. St000068The number of minimal elements in a poset. St000822The Hadwiger number of the graph. St001642The Prague dimension of a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001638The book thickness of a graph. St000741The Colin de Verdière graph invariant.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!