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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> 1
[1,-2] => [1]
=> []
=> 0
[-1,2] => [1]
=> []
=> 0
[2,1] => [2]
=> []
=> 0
[-2,-1] => [2]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> 1
[1,-2,3] => [1,1]
=> [1]
=> 1
[1,-2,-3] => [1]
=> []
=> 0
[-1,2,3] => [1,1]
=> [1]
=> 1
[-1,2,-3] => [1]
=> []
=> 0
[-1,-2,3] => [1]
=> []
=> 0
[1,3,2] => [2,1]
=> [1]
=> 1
[1,3,-2] => [1]
=> []
=> 0
[1,-3,2] => [1]
=> []
=> 0
[1,-3,-2] => [2,1]
=> [1]
=> 1
[-1,3,2] => [2]
=> []
=> 0
[-1,-3,-2] => [2]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-3] => [2]
=> []
=> 0
[2,-1,3] => [1]
=> []
=> 0
[-2,1,3] => [1]
=> []
=> 0
[-2,-1,3] => [2,1]
=> [1]
=> 1
[-2,-1,-3] => [2]
=> []
=> 0
[2,3,1] => [3]
=> []
=> 0
[2,-3,-1] => [3]
=> []
=> 0
[-2,3,-1] => [3]
=> []
=> 0
[-2,-3,1] => [3]
=> []
=> 0
[3,1,2] => [3]
=> []
=> 0
[3,-1,-2] => [3]
=> []
=> 0
[-3,1,-2] => [3]
=> []
=> 0
[-3,-1,2] => [3]
=> []
=> 0
[3,2,1] => [2,1]
=> [1]
=> 1
[3,2,-1] => [1]
=> []
=> 0
[3,-2,1] => [2]
=> []
=> 0
[-3,2,1] => [1]
=> []
=> 0
[-3,2,-1] => [2,1]
=> [1]
=> 1
[-3,-2,-1] => [2]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,-4] => [1,1]
=> [1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,-4] => [1,1]
=> [1]
=> 1
[1,-2,-3,4] => [1,1]
=> [1]
=> 1
[1,-2,-3,-4] => [1]
=> []
=> 0
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,-4] => [1,1]
=> [1]
=> 1
[-1,2,-3,4] => [1,1]
=> [1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St001280
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 0
[1,2] => [1,1]
=> [2]
=> 1
[1,-2] => [1]
=> [1]
=> 0
[-1,2] => [1]
=> [1]
=> 0
[2,1] => [2]
=> [1,1]
=> 0
[-2,-1] => [2]
=> [1,1]
=> 0
[1,2,3] => [1,1,1]
=> [3]
=> 1
[1,2,-3] => [1,1]
=> [2]
=> 1
[1,-2,3] => [1,1]
=> [2]
=> 1
[1,-2,-3] => [1]
=> [1]
=> 0
[-1,2,3] => [1,1]
=> [2]
=> 1
[-1,2,-3] => [1]
=> [1]
=> 0
[-1,-2,3] => [1]
=> [1]
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> 1
[1,3,-2] => [1]
=> [1]
=> 0
[1,-3,2] => [1]
=> [1]
=> 0
[1,-3,-2] => [2,1]
=> [2,1]
=> 1
[-1,3,2] => [2]
=> [1,1]
=> 0
[-1,-3,-2] => [2]
=> [1,1]
=> 0
[2,1,3] => [2,1]
=> [2,1]
=> 1
[2,1,-3] => [2]
=> [1,1]
=> 0
[2,-1,3] => [1]
=> [1]
=> 0
[-2,1,3] => [1]
=> [1]
=> 0
[-2,-1,3] => [2,1]
=> [2,1]
=> 1
[-2,-1,-3] => [2]
=> [1,1]
=> 0
[2,3,1] => [3]
=> [1,1,1]
=> 0
[2,-3,-1] => [3]
=> [1,1,1]
=> 0
[-2,3,-1] => [3]
=> [1,1,1]
=> 0
[-2,-3,1] => [3]
=> [1,1,1]
=> 0
[3,1,2] => [3]
=> [1,1,1]
=> 0
[3,-1,-2] => [3]
=> [1,1,1]
=> 0
[-3,1,-2] => [3]
=> [1,1,1]
=> 0
[-3,-1,2] => [3]
=> [1,1,1]
=> 0
[3,2,1] => [2,1]
=> [2,1]
=> 1
[3,2,-1] => [1]
=> [1]
=> 0
[3,-2,1] => [2]
=> [1,1]
=> 0
[-3,2,1] => [1]
=> [1]
=> 0
[-3,2,-1] => [2,1]
=> [2,1]
=> 1
[-3,-2,-1] => [2]
=> [1,1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,3,-4] => [1,1,1]
=> [3]
=> 1
[1,2,-3,4] => [1,1,1]
=> [3]
=> 1
[1,2,-3,-4] => [1,1]
=> [2]
=> 1
[1,-2,3,4] => [1,1,1]
=> [3]
=> 1
[1,-2,3,-4] => [1,1]
=> [2]
=> 1
[1,-2,-3,4] => [1,1]
=> [2]
=> 1
[1,-2,-3,-4] => [1]
=> [1]
=> 0
[-1,2,3,4] => [1,1,1]
=> [3]
=> 1
[-1,2,3,-4] => [1,1]
=> [2]
=> 1
[-1,2,-3,4] => [1,1]
=> [2]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> [1]
=> 1
[1,-2] => [1]
=> []
=> []
=> 0
[-1,2] => [1]
=> []
=> []
=> 0
[2,1] => [2]
=> []
=> []
=> 0
[-2,-1] => [2]
=> []
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> [1]
=> 1
[1,-2,3] => [1,1]
=> [1]
=> [1]
=> 1
[1,-2,-3] => [1]
=> []
=> []
=> 0
[-1,2,3] => [1,1]
=> [1]
=> [1]
=> 1
[-1,2,-3] => [1]
=> []
=> []
=> 0
[-1,-2,3] => [1]
=> []
=> []
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 1
[1,3,-2] => [1]
=> []
=> []
=> 0
[1,-3,2] => [1]
=> []
=> []
=> 0
[1,-3,-2] => [2,1]
=> [1]
=> [1]
=> 1
[-1,3,2] => [2]
=> []
=> []
=> 0
[-1,-3,-2] => [2]
=> []
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 1
[2,1,-3] => [2]
=> []
=> []
=> 0
[2,-1,3] => [1]
=> []
=> []
=> 0
[-2,1,3] => [1]
=> []
=> []
=> 0
[-2,-1,3] => [2,1]
=> [1]
=> [1]
=> 1
[-2,-1,-3] => [2]
=> []
=> []
=> 0
[2,3,1] => [3]
=> []
=> []
=> 0
[2,-3,-1] => [3]
=> []
=> []
=> 0
[-2,3,-1] => [3]
=> []
=> []
=> 0
[-2,-3,1] => [3]
=> []
=> []
=> 0
[3,1,2] => [3]
=> []
=> []
=> 0
[3,-1,-2] => [3]
=> []
=> []
=> 0
[-3,1,-2] => [3]
=> []
=> []
=> 0
[-3,-1,2] => [3]
=> []
=> []
=> 0
[3,2,1] => [2,1]
=> [1]
=> [1]
=> 1
[3,2,-1] => [1]
=> []
=> []
=> 0
[3,-2,1] => [2]
=> []
=> []
=> 0
[-3,2,1] => [1]
=> []
=> []
=> 0
[-3,2,-1] => [2,1]
=> [1]
=> [1]
=> 1
[-3,-2,-1] => [2]
=> []
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,-3,-4] => [1,1]
=> [1]
=> [1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,-2,3,-4] => [1,1]
=> [1]
=> [1]
=> 1
[1,-2,-3,4] => [1,1]
=> [1]
=> [1]
=> 1
[1,-2,-3,-4] => [1]
=> []
=> []
=> 0
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[-1,2,3,-4] => [1,1]
=> [1]
=> [1]
=> 1
[-1,2,-3,4] => [1,1]
=> [1]
=> [1]
=> 1
Description
The length of the partition.
Matching statistic: St000028
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2] => [1]
=> [[1]]
=> [1] => 0
[-1,2] => [1]
=> [[1]]
=> [1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[-1,3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,-3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 0
[3,-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,2,1] => [1]
=> [[1]]
=> [1] => 0
[-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St000374
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2] => [1]
=> [[1]]
=> [1] => 0
[-1,2] => [1]
=> [[1]]
=> [1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[-1,3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,-3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 0
[3,-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,2,1] => [1]
=> [[1]]
=> [1] => 0
[-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000676
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,-2] => [1]
=> []
=> []
=> 0
[-1,2] => [1]
=> []
=> []
=> 0
[2,1] => [2]
=> []
=> []
=> 0
[-2,-1] => [2]
=> []
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,-2,3] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,-2,-3] => [1]
=> []
=> []
=> 0
[-1,2,3] => [1,1]
=> [1]
=> [1,0]
=> 1
[-1,2,-3] => [1]
=> []
=> []
=> 0
[-1,-2,3] => [1]
=> []
=> []
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 1
[1,3,-2] => [1]
=> []
=> []
=> 0
[1,-3,2] => [1]
=> []
=> []
=> 0
[1,-3,-2] => [2,1]
=> [1]
=> [1,0]
=> 1
[-1,3,2] => [2]
=> []
=> []
=> 0
[-1,-3,-2] => [2]
=> []
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,1,-3] => [2]
=> []
=> []
=> 0
[2,-1,3] => [1]
=> []
=> []
=> 0
[-2,1,3] => [1]
=> []
=> []
=> 0
[-2,-1,3] => [2,1]
=> [1]
=> [1,0]
=> 1
[-2,-1,-3] => [2]
=> []
=> []
=> 0
[2,3,1] => [3]
=> []
=> []
=> 0
[2,-3,-1] => [3]
=> []
=> []
=> 0
[-2,3,-1] => [3]
=> []
=> []
=> 0
[-2,-3,1] => [3]
=> []
=> []
=> 0
[3,1,2] => [3]
=> []
=> []
=> 0
[3,-1,-2] => [3]
=> []
=> []
=> 0
[-3,1,-2] => [3]
=> []
=> []
=> 0
[-3,-1,2] => [3]
=> []
=> []
=> 0
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> 1
[3,2,-1] => [1]
=> []
=> []
=> 0
[3,-2,1] => [2]
=> []
=> []
=> 0
[-3,2,1] => [1]
=> []
=> []
=> 0
[-3,2,-1] => [2,1]
=> [1]
=> [1,0]
=> 1
[-3,-2,-1] => [2]
=> []
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,-3,-4] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,-4] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,-2,-3,-4] => [1]
=> []
=> []
=> 0
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1]
=> [1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1]
=> [1,0]
=> 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000996
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2] => [1]
=> [[1]]
=> [1] => 0
[-1,2] => [1]
=> [[1]]
=> [1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[-1,3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,-3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 0
[3,-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,2,1] => [1]
=> [[1]]
=> [1] => 0
[-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000451
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,-3,2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,-3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[-1,3,2] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[-1,-3,-2] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[2,1,-3] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,-1,3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-2,1,3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-2,-1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[3,-2,1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[-3,2,1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-3,2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2 = 1 + 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001192
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,3,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-3,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[-1,3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-1,-3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,-3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,-1,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-2,1,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,-1] => [1]
=> [1,0]
=> [1,0]
=> 0
[3,-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-3,2,1] => [1]
=> [1,0]
=> [1,0]
=> 0
[-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[-3,-2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,-4] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,5,3,7,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,7,3,4,5,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,5,7,2,3,4,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,6,7,2,5,3,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,6,7,4,2,3,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,1,7,2,4,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,7,5,2,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,1,2,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,1,6,2,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,5,1,2,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,5,6,1,2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,4,6,1,2,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,5,1,2,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,7,5,6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,3,7,1,2,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,2,5,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,3,2,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,1,4,2,3,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,2,6,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,5,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,5,6,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,6,2,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,6,4,5,2,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,4,1,3,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,2,6,1,3,4,5] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,3,1,2,4,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,5,6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,6,1,2,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,3,1,2,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[5,6,7,8,4,1,2,3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,1,4,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[6,8,-4,-3,1,2,5,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[-3,2,-8,-6,-4,1,5,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,7,-4,2,3,6,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[-4,5,2,3,8,-7,1,6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-3,6,8,-5,1,2,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[4,3,-5,2,8,-7,1,6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-4,3,-7,-5,2,6,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[4,3,7,-6,2,5,-8,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,-4,-8,-6,2,-7,1,5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,4,-7,-6,3,1,2,5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,-3,4,8,-7,1,2,6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-5,2,4,8,-7,1,3,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[4,-7,-5,6,-8,-3,1,2] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-6,-4,3,5,-7,2,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2
[8,-6,2,4,-7,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[-7,6,8,1,3,-5,2,4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001215
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001215: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001215: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,3,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-3,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[-1,3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-1,-3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,-3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,-1,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-2,1,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-2,-1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,-1] => [1]
=> [1,0]
=> [1,0]
=> 0
[3,-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-3,2,1] => [1]
=> [1,0]
=> [1,0]
=> 0
[-3,2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[-3,-2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,-4] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,5,3,7,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,7,3,4,5,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,5,7,2,3,4,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,6,7,2,5,3,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,6,7,4,2,3,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,1,7,2,4,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,7,5,2,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,1,2,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,1,6,2,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,5,1,2,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,5,6,1,2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,4,6,1,2,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,5,1,2,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,7,5,6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,3,7,1,2,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,2,5,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,3,2,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,1,4,2,3,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,2,6,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,5,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,5,6,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,6,2,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,6,4,5,2,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,4,1,3,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,2,6,1,3,4,5] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,3,1,2,4,5,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,5,6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,6,1,2,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,3,1,2,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[5,6,7,8,4,1,2,3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,1,4,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[6,8,-4,-3,1,2,5,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[-3,2,-8,-6,-4,1,5,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,7,-4,2,3,6,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[-4,5,2,3,8,-7,1,6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-3,6,8,-5,1,2,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[4,3,-5,2,8,-7,1,6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-4,3,-7,-5,2,6,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[4,3,7,-6,2,5,-8,1] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,-4,-8,-6,2,-7,1,5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,4,-7,-6,3,1,2,5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,-3,4,8,-7,1,2,6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-5,2,4,8,-7,1,3,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[4,-7,-5,6,-8,-3,1,2] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[-6,-4,3,5,-7,2,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2
[8,-6,2,4,-7,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[-7,6,8,1,3,-5,2,4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module.
For the first 196 values, the statistic also gives the number of indecomposable non-projective modules $X$ such that $\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001557The number of inversions of the second entry of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000734The last entry in the first row of a standard tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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