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Your data matches 82 different statistics following compositions of up to 3 maps.
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Matching statistic: St000145
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 1 = 2 - 1
[1,1]
=> -1 = 0 - 1
[3]
=> 2 = 3 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> -2 = -1 - 1
[4]
=> 3 = 4 - 1
[3,1]
=> 1 = 2 - 1
[2,2]
=> 0 = 1 - 1
[2,1,1]
=> -1 = 0 - 1
[1,1,1,1]
=> -3 = -2 - 1
[5]
=> 4 = 5 - 1
[4,1]
=> 2 = 3 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 0 = 1 - 1
[2,2,1]
=> -1 = 0 - 1
[2,1,1,1]
=> -2 = -1 - 1
[1,1,1,1,1]
=> -4 = -3 - 1
[6]
=> 5 = 6 - 1
[5,1]
=> 3 = 4 - 1
[4,2]
=> 2 = 3 - 1
[4,1,1]
=> 1 = 2 - 1
[3,3]
=> 1 = 2 - 1
[3,2,1]
=> 0 = 1 - 1
[3,1,1,1]
=> -1 = 0 - 1
[2,2,2]
=> -1 = 0 - 1
[2,2,1,1]
=> -2 = -1 - 1
[2,1,1,1,1]
=> -3 = -2 - 1
[1,1,1,1,1,1]
=> -5 = -4 - 1
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000878
(load all 86 compositions to match this statistic)
(load all 86 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => -1 = 0 - 1
[1,1]
=> 110 => 1 = 2 - 1
[3]
=> 1000 => -2 = -1 - 1
[2,1]
=> 1010 => 0 = 1 - 1
[1,1,1]
=> 1110 => 2 = 3 - 1
[4]
=> 10000 => -3 = -2 - 1
[3,1]
=> 10010 => -1 = 0 - 1
[2,2]
=> 1100 => 0 = 1 - 1
[2,1,1]
=> 10110 => 1 = 2 - 1
[1,1,1,1]
=> 11110 => 3 = 4 - 1
[5]
=> 100000 => -4 = -3 - 1
[4,1]
=> 100010 => -2 = -1 - 1
[3,2]
=> 10100 => -1 = 0 - 1
[3,1,1]
=> 100110 => 0 = 1 - 1
[2,2,1]
=> 11010 => 1 = 2 - 1
[2,1,1,1]
=> 101110 => 2 = 3 - 1
[1,1,1,1,1]
=> 111110 => 4 = 5 - 1
[6]
=> 1000000 => -5 = -4 - 1
[5,1]
=> 1000010 => -3 = -2 - 1
[4,2]
=> 100100 => -2 = -1 - 1
[4,1,1]
=> 1000110 => -1 = 0 - 1
[3,3]
=> 11000 => -1 = 0 - 1
[3,2,1]
=> 101010 => 0 = 1 - 1
[3,1,1,1]
=> 1001110 => 1 = 2 - 1
[2,2,2]
=> 11100 => 1 = 2 - 1
[2,2,1,1]
=> 110110 => 2 = 3 - 1
[2,1,1,1,1]
=> 1011110 => 3 = 4 - 1
[1,1,1,1,1,1]
=> 1111110 => 5 = 6 - 1
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000894
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> -1
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> -2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> -1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> -3
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> -1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> -2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> -4
Description
The trace of an alternating sign matrix.
Matching statistic: St000090
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => [2,1] => -1 = 0 - 1
[1,1]
=> 110 => 011 => [1,2] => 1 = 2 - 1
[3]
=> 1000 => 0001 => [3,1] => -2 = -1 - 1
[2,1]
=> 1010 => 0011 => [2,2] => 0 = 1 - 1
[1,1,1]
=> 1110 => 0111 => [1,3] => 2 = 3 - 1
[4]
=> 10000 => 00001 => [4,1] => -3 = -2 - 1
[3,1]
=> 10010 => 00011 => [3,2] => -1 = 0 - 1
[2,2]
=> 1100 => 0011 => [2,2] => 0 = 1 - 1
[2,1,1]
=> 10110 => 00111 => [2,3] => 1 = 2 - 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 3 = 4 - 1
[5]
=> 100000 => 000001 => [5,1] => -4 = -3 - 1
[4,1]
=> 100010 => 000011 => [4,2] => -2 = -1 - 1
[3,2]
=> 10100 => 00011 => [3,2] => -1 = 0 - 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 0 = 1 - 1
[2,2,1]
=> 11010 => 00111 => [2,3] => 1 = 2 - 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 2 = 3 - 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 4 = 5 - 1
[6]
=> 1000000 => 0000001 => [6,1] => -5 = -4 - 1
[5,1]
=> 1000010 => 0000011 => [5,2] => -3 = -2 - 1
[4,2]
=> 100100 => 000011 => [4,2] => -2 = -1 - 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => -1 = 0 - 1
[3,3]
=> 11000 => 00011 => [3,2] => -1 = 0 - 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 0 = 1 - 1
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1 = 2 - 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1 = 2 - 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 2 = 3 - 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 3 = 4 - 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 5 = 6 - 1
Description
The variation of a composition.
Matching statistic: St001232
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = -1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {-2,0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {-2,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {-2,0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {-3,-1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {-3,-1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {-3,-1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {-4,-2,-1,0,0,2,2}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000993
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Values
[2]
=> []
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ? ∊ {-1,1,3}
[2,1]
=> [1]
=> []
=> ? ∊ {-1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-1,1,3}
[4]
=> []
=> ?
=> ? ∊ {-2,0,1,4}
[3,1]
=> [1]
=> []
=> ? ∊ {-2,0,1,4}
[2,2]
=> [2]
=> []
=> ? ∊ {-2,0,1,4}
[2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-2,0,1,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[5]
=> []
=> ?
=> ? ∊ {-3,-1,0,1,5}
[4,1]
=> [1]
=> []
=> ? ∊ {-3,-1,0,1,5}
[3,2]
=> [2]
=> []
=> ? ∊ {-3,-1,0,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-3,-1,0,1,5}
[2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-3,-1,0,1,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[6]
=> []
=> ?
=> ? ∊ {-4,-2,-1,0,0,6}
[5,1]
=> [1]
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[4,2]
=> [2]
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[4,1,1]
=> [1,1]
=> [1]
=> ? ∊ {-4,-2,-1,0,0,6}
[3,3]
=> [3]
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {-4,-2,-1,0,0,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000477
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000477: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {-1,1,3}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {-1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {-2,0,1,4}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {-2,0,1,4}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {-2,0,1,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-2,0,1,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[5]
=> []
=> ?
=> ?
=> ? ∊ {-3,-1,1,3,5}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {-3,-1,1,3,5}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {-3,-1,1,3,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,1,3,5}
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,1,3,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {-4,-2,0,1,3,6}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {-4,-2,0,1,3,6}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {-4,-2,0,1,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,0,1,3,6}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {-4,-2,0,1,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,0,1,3,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> -1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
Description
The weight of a partition according to Alladi.
Matching statistic: St000567
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {-1,1,3}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {-1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {-2,1,2,4}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {-2,1,2,4}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {-2,1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-2,1,2,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {-3,-1,1,3,5}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {-3,-1,1,3,5}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {-3,-1,1,3,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,1,3,5}
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,1,3,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
[6]
=> []
=> ?
=> ?
=> ? ∊ {-4,-2,-1,2,3,6}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {-4,-2,-1,2,3,6}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {-4,-2,-1,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,-1,2,3,6}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {-4,-2,-1,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,-1,2,3,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 4
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {-1,1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {-1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {-2,0,1,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {-2,0,1,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {-2,0,1,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-2,0,1,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[5]
=> []
=> ?
=> ?
=> ? ∊ {-3,-1,0,1,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {-3,-1,0,1,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {-3,-1,0,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,0,1,5}
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,0,1,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[6]
=> []
=> ?
=> ?
=> ? ∊ {-4,-2,-1,0,0,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,-1,0,0,6}
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,-1,0,0,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 36%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {-1,1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {-1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {-2,0,1,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {-2,0,1,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {-2,0,1,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-2,0,1,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[5]
=> []
=> ?
=> ?
=> ? ∊ {-3,-1,0,1,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {-3,-1,0,1,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {-3,-1,0,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,0,1,5}
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-3,-1,0,1,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[6]
=> []
=> ?
=> ?
=> ? ∊ {-4,-2,-1,0,0,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,-1,0,0,6}
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {-4,-2,-1,0,0,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {-4,-2,-1,0,0,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
Description
The product of the parts of an integer partition.
The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000997The even-odd crank of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000941The number of characters of the symmetric group whose value on the partition is even. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000707The product of the factorials of the parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001823The Stasinski-Voll length of a signed permutation. St001842The major index of a set partition. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St001209The pmaj statistic of a parking function. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001415The length of the longest palindromic prefix of a binary word. St000868The aid statistic in the sense of Shareshian-Wachs. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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