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Matching statistic: St001232
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 0
[2,1] => 00 => [2] => [1,1,0,0]
=> 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,2,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,-2,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-2,-3,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001227
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 0
[2,1] => 00 => [2] => [1,1,0,0]
=> 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,2,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[-3,-2,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,-2,-3,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,7,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,5,3,7,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,7,3,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,7,3,6,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,4,7,2,3,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,5,7,2,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,6,7,2,3,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,6,7,2,5,3,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,6,7,4,2,3,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,7,2,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,7,5,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,1,2,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,1,6,2,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,5,1,2,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,7,5,6,1,2] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,5,7,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,6,1,2,7,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,4,1,2,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,4,5,1,2,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,4,5,6,1,2] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,4,6,1,2,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,5,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,5,6,1,2,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,7,6,1,2,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,7,5,6,1,2,3] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,3,4,7,1,2,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,3,7,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,4,7,1,2,3,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,2,5,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,3,2,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,2,3,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,2,6,3,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,5,2,3,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,4,6,2,3,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,5,2,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,5,6,2,3,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,6,2,3,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,1,6,4,5,2,3] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,2,4,1,3,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,2,5,1,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,2,6,1,3,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,1,2,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,5,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,5,6,1,2,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,3,6,1,2,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,5,1,4,2,3,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-1,2,3,4,5,6,7] => 1000000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,2,-7,-6,3,4,5] => 0011000 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,7,-6,-5,2,3,4] => 0011000 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,5,-7,-4,2,3,6] => 0011000 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
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