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Matching statistic: St001563
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St001563: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 4
[3]
=> 1
[2,1]
=> 4
[1,1,1]
=> 27
[4]
=> 1
[3,1]
=> 4
[2,2]
=> 4
[2,1,1]
=> 27
[1,1,1,1]
=> 256
[5]
=> 1
[4,1]
=> 4
[3,2]
=> 4
[3,1,1]
=> 27
[2,2,1]
=> 27
[2,1,1,1]
=> 256
[1,1,1,1,1]
=> 3125
[6]
=> 1
[5,1]
=> 4
[4,2]
=> 4
[4,1,1]
=> 27
[3,3]
=> 4
[3,2,1]
=> 27
[3,1,1,1]
=> 256
[2,2,2]
=> 27
[2,2,1,1]
=> 256
[2,1,1,1,1]
=> 3125
[1,1,1,1,1,1]
=> 46656
[7]
=> 1
[6,1]
=> 4
[5,2]
=> 4
[5,1,1]
=> 27
[4,3]
=> 4
[4,2,1]
=> 27
[4,1,1,1]
=> 256
[3,3,1]
=> 27
[3,2,2]
=> 27
[3,2,1,1]
=> 256
[3,1,1,1,1]
=> 3125
[2,2,2,1]
=> 256
[2,2,1,1,1]
=> 3125
[2,1,1,1,1,1]
=> 46656
[1,1,1,1,1,1,1]
=> 823543
Description
The value of the power-sum symmetric function evaluated at 1.
The statistic is $p_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$,
where $\lambda$ has $k$ parts.
Matching statistic: St001632
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 14%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 14%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> ? = 1
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 4
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 4
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ? = 27
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 27
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ? = 256
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? = 4
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? = 27
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? = 27
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ? = 256
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> ? = 3125
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? = 4
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 4
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> ? = 27
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 4
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> ? = 27
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> ? = 256
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ? = 27
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ? = 256
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> ? = 3125
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> ? = 46656
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 4
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 27
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 27
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> ? = 256
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 27
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 27
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 256
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
=> ? = 3125
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 256
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
=> ? = 3125
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(5,6)],7)
=> ? = 46656
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([],7)
=> ? = 823543
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001491
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 14%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 14%
Values
[1]
=> []
=> []
=> => ? = 1 - 4
[2]
=> []
=> []
=> => ? = 1 - 4
[1,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 4 - 4
[3]
=> []
=> []
=> => ? = 1 - 4
[2,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 4 - 4
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 27 - 4
[4]
=> []
=> []
=> => ? = 1 - 4
[3,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 4 - 4
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 4 - 4
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 27 - 4
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 256 - 4
[5]
=> []
=> []
=> => ? = 1 - 4
[4,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 4 - 4
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 4 - 4
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 27 - 4
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 27 - 4
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 256 - 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3125 - 4
[6]
=> []
=> []
=> => ? = 1 - 4
[5,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 4 - 4
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 4 - 4
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 27 - 4
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 4 - 4
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 27 - 4
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 256 - 4
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 27 - 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 256 - 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3125 - 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 46656 - 4
[7]
=> []
=> []
=> => ? = 1 - 4
[6,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 4 - 4
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 4 - 4
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 27 - 4
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 4 - 4
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 27 - 4
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 256 - 4
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 27 - 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 27 - 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 256 - 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3125 - 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 256 - 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 3125 - 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 46656 - 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 823543 - 4
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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