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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St001414
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(load all 20 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0
[1,1] => 11 => 0
[2] => 10 => 0
[1,1,1] => 111 => 1
[1,2] => 110 => 0
[2,1] => 101 => 1
[3] => 100 => 0
[1,1,1,1] => 1111 => 1
[1,1,2] => 1110 => 1
[1,2,1] => 1101 => 0
[1,3] => 1100 => 0
[2,1,1] => 1011 => 1
[2,2] => 1010 => 1
[3,1] => 1001 => 0
[4] => 1000 => 0
[1,1,1,1,1] => 11111 => 2
[1,1,1,2] => 11110 => 1
[1,1,2,1] => 11101 => 1
[1,1,3] => 11100 => 1
[1,2,1,1] => 11011 => 2
[1,2,2] => 11010 => 0
[1,3,1] => 11001 => 0
[1,4] => 11000 => 0
[2,1,1,1] => 10111 => 1
[2,1,2] => 10110 => 1
[2,2,1] => 10101 => 2
[2,3] => 10100 => 1
[3,1,1] => 10011 => 0
[3,2] => 10010 => 0
[4,1] => 10001 => 2
[5] => 10000 => 0
[1,1,1,1,1,1] => 111111 => 2
[1,1,1,1,2] => 111110 => 2
[1,1,1,2,1] => 111101 => 1
[1,1,1,3] => 111100 => 1
[1,1,2,1,1] => 111011 => 1
[1,1,2,2] => 111010 => 1
[1,1,3,1] => 111001 => 1
[1,1,4] => 111000 => 1
[1,2,1,1,1] => 110111 => 2
[1,2,1,2] => 110110 => 2
[1,2,2,1] => 110101 => 0
[1,2,3] => 110100 => 0
[1,3,1,1] => 110011 => 0
[1,3,2] => 110010 => 0
[1,4,1] => 110001 => 0
[1,5] => 110000 => 0
[2,1,1,1,1] => 101111 => 1
[2,1,1,2] => 101110 => 1
[2,1,2,1] => 101101 => 1
Description
Half the length of the longest odd length palindromic prefix of a binary word.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
Matching statistic: St001413
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 => 0 => 0
[1,1] => 11 => 00 => 01 => 0
[2] => 10 => 01 => 10 => 0
[1,1,1] => 111 => 000 => 011 => 0
[1,2] => 110 => 001 => 101 => 0
[2,1] => 101 => 010 => 000 => 1
[3] => 100 => 011 => 110 => 1
[1,1,1,1] => 1111 => 0000 => 0111 => 0
[1,1,2] => 1110 => 0001 => 1011 => 0
[1,2,1] => 1101 => 0010 => 0001 => 1
[1,3] => 1100 => 0011 => 1101 => 1
[2,1,1] => 1011 => 0100 => 0100 => 0
[2,2] => 1010 => 0101 => 1000 => 0
[3,1] => 1001 => 0110 => 0010 => 1
[4] => 1000 => 0111 => 1110 => 1
[1,1,1,1,1] => 11111 => 00000 => 01111 => 0
[1,1,1,2] => 11110 => 00001 => 10111 => 0
[1,1,2,1] => 11101 => 00010 => 00011 => 1
[1,1,3] => 11100 => 00011 => 11011 => 1
[1,2,1,1] => 11011 => 00100 => 01001 => 0
[1,2,2] => 11010 => 00101 => 10001 => 0
[1,3,1] => 11001 => 00110 => 00101 => 1
[1,4] => 11000 => 00111 => 11101 => 1
[2,1,1,1] => 10111 => 01000 => 01100 => 2
[2,1,2] => 10110 => 01001 => 10100 => 0
[2,2,1] => 10101 => 01010 => 00000 => 2
[2,3] => 10100 => 01011 => 11000 => 1
[3,1,1] => 10011 => 01100 => 01010 => 0
[3,2] => 10010 => 01101 => 10010 => 2
[4,1] => 10001 => 01110 => 00110 => 1
[5] => 10000 => 01111 => 11110 => 2
[1,1,1,1,1,1] => 111111 => 000000 => 011111 => 0
[1,1,1,1,2] => 111110 => 000001 => 101111 => 0
[1,1,1,2,1] => 111101 => 000010 => 000111 => 1
[1,1,1,3] => 111100 => 000011 => 110111 => 1
[1,1,2,1,1] => 111011 => 000100 => 010011 => 0
[1,1,2,2] => 111010 => 000101 => 100011 => 0
[1,1,3,1] => 111001 => 000110 => 001011 => 1
[1,1,4] => 111000 => 000111 => 111011 => 1
[1,2,1,1,1] => 110111 => 001000 => 011001 => 2
[1,2,1,2] => 110110 => 001001 => 101001 => 0
[1,2,2,1] => 110101 => 001010 => 000001 => 2
[1,2,3] => 110100 => 001011 => 110001 => 1
[1,3,1,1] => 110011 => 001100 => 010101 => 0
[1,3,2] => 110010 => 001101 => 100101 => 2
[1,4,1] => 110001 => 001110 => 001101 => 1
[1,5] => 110000 => 001111 => 111101 => 2
[2,1,1,1,1] => 101111 => 010000 => 011100 => 0
[2,1,1,2] => 101110 => 010001 => 101100 => 0
[2,1,2,1] => 101101 => 010010 => 000100 => 1
Description
Half the length of the longest even length palindromic prefix of a binary word.
Matching statistic: St000777
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1} + 1
[2,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1} + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,1,1} + 1
[1,3] => [1,1] => [2] => ([],2)
=> ? ∊ {0,1,1} + 1
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,2] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[3,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,1,1} + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[1,4] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,3] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,2] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[4,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,2,2,2} + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,2,2] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,1,3,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,4,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[1,5] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[2,1,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[2,2,1,1] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[2,2,2] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[2,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[2,4] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[3,3] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,2] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[5,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2} + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,2,1] => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,3,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,2,1,2] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,2,3] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,3,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,4,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,1,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,2,1,3] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,2,2,1,1] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,2,4] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,3,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,5,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,6] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[2,1,3,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[2,1,4] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[2,2,1,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[2,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000137
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 80%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St001280
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001283
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
Description
The number of finite groups that are realised by the given partition over the complex numbers.
A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St001629
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,1] => [2] => [1,1] => [2] => ? ∊ {0,0}
[2] => [1] => [1] => [1] => ? ∊ {0,0}
[1,1,1] => [3] => [1,1,1] => [3] => 1
[1,2] => [1,1] => [2] => [1] => ? ∊ {0,0,1}
[2,1] => [1,1] => [2] => [1] => ? ∊ {0,0,1}
[3] => [1] => [1] => [1] => ? ∊ {0,0,1}
[1,1,1,1] => [4] => [1,1,1,1] => [4] => 1
[1,1,2] => [2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,1,1,1}
[1,2,1] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,1,1,1}
[1,3] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,1,1,1}
[2,1,1] => [1,2] => [1,2] => [1,1] => ? ∊ {0,0,0,0,1,1,1}
[2,2] => [2] => [1,1] => [2] => ? ∊ {0,0,0,0,1,1,1}
[3,1] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,1,1,1}
[4] => [1] => [1] => [1] => ? ∊ {0,0,0,0,1,1,1}
[1,1,1,1,1] => [5] => [1,1,1,1,1] => [5] => 1
[1,1,1,2] => [3,1] => [2,1,1] => [1,2] => 0
[1,1,2,1] => [2,1,1] => [3,1] => [1,1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,1,3] => [2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,1,1] => [1,1,2] => [1,3] => [1,1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [1,2] => [1,1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [1,1,2] => [2,1] => 0
[2,1,2] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,3] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [1,2] => [1,1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,2] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[5] => [1] => [1] => [1] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => [6] => 1
[1,1,1,1,2] => [4,1] => [2,1,1,1] => [1,3] => 0
[1,1,1,2,1] => [3,1,1] => [3,1,1] => [1,2] => 0
[1,1,1,3] => [3,1] => [2,1,1] => [1,2] => 0
[1,1,2,1,1] => [2,1,2] => [1,3,1] => [1,1,1] => 1
[1,1,2,2] => [2,2] => [1,2,1] => [1,1,1] => 1
[1,1,3,1] => [2,1,1] => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4] => [2,1] => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [1,1,3] => [1,1,3] => [2,1] => 0
[1,2,1,2] => [1,1,1,1] => [4] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [1,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [1,1,1,2] => [3,1] => 0
[2,1,1,2] => [1,2,1] => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1] => [1,1,1,1] => [4] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [1,2,1] => [1,1,1] => 1
[2,2,2] => [3] => [1,1,1] => [3] => 1
[2,3,1] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [1,1,2] => [2,1] => 0
[3,1,2] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1] => [1,1,1] => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3] => [2] => [1,1] => [2] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,1] => [1,2] => [1,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,2] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1] => [1,1] => [2] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6] => [1] => [1] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => [7] => 1
[1,1,1,1,1,2] => [5,1] => [2,1,1,1,1] => [1,4] => 0
[1,1,1,1,2,1] => [4,1,1] => [3,1,1,1] => [1,3] => 0
[1,1,1,1,3] => [4,1] => [2,1,1,1] => [1,3] => 0
[1,1,1,2,1,1] => [3,1,2] => [1,3,1,1] => [1,1,2] => 1
[1,1,1,2,2] => [3,2] => [1,2,1,1] => [1,1,2] => 1
[1,1,1,3,1] => [3,1,1] => [3,1,1] => [1,2] => 0
[1,1,1,4] => [3,1] => [2,1,1] => [1,2] => 0
[1,1,2,1,1,1] => [2,1,3] => [1,1,3,1] => [2,1,1] => 1
[1,1,2,1,2] => [2,1,1,1] => [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,2,2,1] => [2,2,1] => [2,2,1] => [2,1] => 0
[1,1,2,3] => [2,1,1] => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,3,1,1] => [2,1,2] => [1,3,1] => [1,1,1] => 1
[1,1,3,2] => [2,1,1] => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,1,1] => [1,1,4] => [1,1,1,3] => [3,1] => 0
[1,2,2,1,1] => [1,2,2] => [1,2,2] => [1,2] => 0
[1,2,2,2] => [1,3] => [1,1,2] => [2,1] => 0
[1,3,1,1,1] => [1,1,3] => [1,1,3] => [2,1] => 0
[2,1,1,1,1,1] => [1,5] => [1,1,1,1,2] => [4,1] => 0
[2,1,1,1,2] => [1,3,1] => [2,1,2] => [1,1,1] => 1
[2,2,1,1,1] => [2,3] => [1,1,2,1] => [2,1,1] => 1
[2,2,2,1] => [3,1] => [2,1,1] => [1,2] => 0
[3,1,1,1,1] => [1,4] => [1,1,1,2] => [3,1] => 0
[4,1,1,1] => [1,3] => [1,1,2] => [2,1] => 0
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1] => [8] => 1
[1,1,1,1,1,1,2] => [6,1] => [2,1,1,1,1,1] => [1,5] => 0
[1,1,1,1,1,2,1] => [5,1,1] => [3,1,1,1,1] => [1,4] => 0
[1,1,1,1,1,3] => [5,1] => [2,1,1,1,1] => [1,4] => 0
[1,1,1,1,2,1,1] => [4,1,2] => [1,3,1,1,1] => [1,1,3] => 2
[1,1,1,1,2,2] => [4,2] => [1,2,1,1,1] => [1,1,3] => 2
[1,1,1,1,3,1] => [4,1,1] => [3,1,1,1] => [1,3] => 0
[1,1,1,1,4] => [4,1] => [2,1,1,1] => [1,3] => 0
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,3,1,1] => [2,1,2] => 3
[1,1,1,2,1,2] => [3,1,1,1] => [4,1,1] => [1,2] => 0
[1,1,1,2,2,1] => [3,2,1] => [2,2,1,1] => [2,2] => 1
[1,1,1,2,3] => [3,1,1] => [3,1,1] => [1,2] => 0
[1,1,1,3,1,1] => [3,1,2] => [1,3,1,1] => [1,1,2] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000929
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 0
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 0
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001123
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001568The smallest positive integer that does not appear twice in the partition. St001651The Frankl number of a lattice. St001115The number of even descents of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001877Number of indecomposable injective modules with projective dimension 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. 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. 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. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000901The cube of the number of standard Young tableaux with shape given by the 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral.
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