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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001683
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[2,3,4,5,1] => 0
[2,3,5,4,1] => 1
[2,4,3,5,1] => 1
[2,4,5,3,1] => 2
[2,5,3,4,1] => 1
[2,5,4,3,1] => 2
[3,2,4,5,1] => 0
[3,2,5,4,1] => 1
[3,4,2,5,1] => 0
[3,4,5,2,1] => 0
[3,5,2,4,1] => 1
[3,5,4,2,1] => 1
[4,2,3,5,1] => 0
[4,2,5,3,1] => 1
[4,3,2,5,1] => 0
[4,3,5,2,1] => 0
[4,5,2,3,1] => 0
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000356
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => 0
[3,1,2] => [2,3,1] => 0
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,1,2,4] => 0
[2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [4,1,3,2] => 1
[3,1,2,4] => [2,3,1,4] => 0
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => 0
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => 0
[4,1,3,2] => [2,4,3,1] => 1
[4,2,1,3] => [3,2,4,1] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [3,4,2,1] => 0
[4,3,2,1] => [4,3,2,1] => 0
[2,3,4,5,1] => [5,1,2,3,4] => 0
[2,3,5,4,1] => [5,1,2,4,3] => 1
[2,4,3,5,1] => [5,1,3,2,4] => 1
[2,4,5,3,1] => [5,1,4,2,3] => 2
[2,5,3,4,1] => [5,1,3,4,2] => 1
[2,5,4,3,1] => [5,1,4,3,2] => 2
[3,2,4,5,1] => [5,2,1,3,4] => 0
[3,2,5,4,1] => [5,2,1,4,3] => 1
[3,4,2,5,1] => [5,3,1,2,4] => 0
[3,4,5,2,1] => [5,4,1,2,3] => 0
[3,5,2,4,1] => [5,3,1,4,2] => 1
[3,5,4,2,1] => [5,4,1,3,2] => 1
[4,2,3,5,1] => [5,2,3,1,4] => 0
[4,2,5,3,1] => [5,2,4,1,3] => 1
[4,3,2,5,1] => [5,3,2,1,4] => 0
[4,3,5,2,1] => [5,4,2,1,3] => 0
[4,5,2,3,1] => [5,3,4,1,2] => 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000358
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [3,1,2] => [2,3,1] => 0
[2,3,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => 2
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => 1
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 2
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 0
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 0
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,2,3] => 1
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,5,2,4,3] => 2
[2,5,3,4,1] => [1,4,3,5,2] => [1,5,3,2,4] => 1
[2,5,4,3,1] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,4,5,3,2] => 0
[3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [1,3,5,4,2] => 0
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[4,2,3,5,1] => [1,5,3,2,4] => [1,4,3,5,2] => 0
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001882
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 2
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => 1
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [2,3,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,3,5,4,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[2,5,3,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[4,2,3,5,1] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St000223
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [3,1,4,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [2,3,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [3,2,1,4] => 1
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [3,4,1,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 1
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[2,3,5,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[2,4,3,5,1] => [5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[2,4,5,3,1] => [5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[2,5,3,4,1] => [5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[2,5,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[3,2,4,5,1] => [5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[3,2,5,4,1] => [5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[3,4,2,5,1] => [5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[3,4,5,2,1] => [5,4,1,2,3] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[3,5,2,4,1] => [5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[3,5,4,2,1] => [5,4,1,3,2] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[4,2,3,5,1] => [5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => 0
[4,2,5,3,1] => [5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[4,3,2,5,1] => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[4,3,5,2,1] => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[4,5,2,3,1] => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of nestings in the permutation.
Matching statistic: St000371
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [3,1,4,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [2,3,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [3,2,1,4] => 1
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [3,4,1,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 1
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[2,3,5,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[2,4,3,5,1] => [5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[2,4,5,3,1] => [5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[2,5,3,4,1] => [5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[2,5,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[3,2,4,5,1] => [5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[3,2,5,4,1] => [5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[3,4,2,5,1] => [5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[3,4,5,2,1] => [5,4,1,2,3] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[3,5,2,4,1] => [5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[3,5,4,2,1] => [5,4,1,3,2] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[4,2,3,5,1] => [5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => 0
[4,2,5,3,1] => [5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[4,3,2,5,1] => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[4,3,5,2,1] => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[4,5,2,3,1] => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
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: St001687
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => [3,1,2] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [2,3,1] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [2,4,1,3] => 1
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 2
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => [2,1,3,4] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,3,4,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,1,4,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [1,2,4,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,1,4,3] => 1
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [4,2,1,3] => 1
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,3,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 0
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => [4,1,3,2] => 0
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [2,3,4,1] => 0
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => [3,2,4,1] => 1
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [2,4,3,1] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [3,4,2,1] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => [5,1,2,3,4] => 0
[2,3,5,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => [5,3,1,2,4] => 1
[2,4,3,5,1] => [5,1,3,2,4] => [4,2,3,1,5] => [5,2,4,1,3] => 1
[2,4,5,3,1] => [5,1,4,2,3] => [3,2,4,1,5] => [5,2,3,1,4] => 2
[2,5,3,4,1] => [5,1,3,4,2] => [2,4,3,1,5] => [5,2,1,3,4] => 1
[2,5,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => [5,3,2,1,4] => 2
[3,2,4,5,1] => [5,2,1,3,4] => [4,3,1,2,5] => [5,1,3,4,2] => 0
[3,2,5,4,1] => [5,2,1,4,3] => [3,4,1,2,5] => [5,3,1,4,2] => 1
[3,4,2,5,1] => [5,3,1,2,4] => [4,2,1,3,5] => [5,1,2,4,3] => 0
[3,4,5,2,1] => [5,4,1,2,3] => [3,2,1,4,5] => [5,4,1,2,3] => 0
[3,5,2,4,1] => [5,3,1,4,2] => [2,4,1,3,5] => [5,2,1,4,3] => 1
[3,5,4,2,1] => [5,4,1,3,2] => [2,3,1,4,5] => [5,4,2,1,3] => 1
[4,2,3,5,1] => [5,2,3,1,4] => [4,1,3,2,5] => [5,1,4,2,3] => 0
[4,2,5,3,1] => [5,2,4,1,3] => [3,1,4,2,5] => [5,1,3,2,4] => 1
[4,3,2,5,1] => [5,3,2,1,4] => [4,1,2,3,5] => [5,1,4,3,2] => 0
[4,3,5,2,1] => [5,4,2,1,3] => [3,1,2,4,5] => [5,4,1,3,2] => 0
[4,5,2,3,1] => [5,3,4,1,2] => [2,1,4,3,5] => [5,3,4,1,2] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St001232
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 40%●distinct values known / distinct values provided: 33%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 40%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 1
[2,3,1] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[3,4,1,2] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[4,2,3,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[2,4,3,5,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[2,4,5,3,1] => [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[3,2,4,5,1] => [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[3,2,5,4,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[3,4,2,5,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,5,2,4,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[3,5,4,2,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[4,2,3,5,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[4,2,5,3,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[4,3,2,5,1] => [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[4,3,5,2,1] => [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[4,5,2,3,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,2,3,4,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,2,4,3,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[5,3,2,4,1] => [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,4,2,3,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[1,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 0
[2,1,4,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 0
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[3,1,2,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 0
[3,1,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[3,2,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 0
[3,2,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 0
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2
[3,2,4,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 0
[3,2,5,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[3,4,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 0
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[3,5,4,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[4,2,3,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 0
[4,2,5,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[4,3,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 0
[4,3,5,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 0
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 0
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[1,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 0
[2,1,4,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 0
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[3,1,2,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 0
[3,1,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[3,2,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 0
[3,2,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 0
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2
[3,2,4,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 0
[3,2,5,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[3,4,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 0
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[3,5,4,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[4,2,3,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 0
[4,2,5,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[4,3,2,5,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ? = 0
[4,3,5,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 0
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 1
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ? = 0
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001857The number of edges in the reduced word graph of a signed permutation. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000709The number of occurrences of 14-2-3 or 14-3-2. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001868The number of alignments of type NE of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001964The interval resolution global dimension of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000181The number of connected components of the Hasse diagram for the poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001890The maximum magnitude of the Möbius function of a poset. St000264The girth of a graph, which is not a tree.
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