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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000149
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(load all 5 compositions to match this statistic)
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 2
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 2
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 3
[5,1]
=> 2
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 0
[3,1,1,1]
=> 1
[2,2,2]
=> 1
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 3
[6,1]
=> 2
[5,2]
=> 2
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 0
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 0
[2,2,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 4
[7,1]
=> 3
[6,2]
=> 3
[6,1,1]
=> 2
[5,3]
=> 2
[5,2,1]
=> 1
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000288
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Values
[1]
=> [1]
=> 1 => 0 => 0
[2]
=> [2]
=> 0 => 1 => 1
[1,1]
=> [1,1]
=> 11 => 00 => 0
[3]
=> [2,1]
=> 01 => 10 => 1
[2,1]
=> [3]
=> 1 => 0 => 0
[1,1,1]
=> [1,1,1]
=> 111 => 000 => 0
[4]
=> [2,2]
=> 00 => 11 => 2
[3,1]
=> [2,1,1]
=> 011 => 100 => 1
[2,2]
=> [4]
=> 0 => 1 => 1
[2,1,1]
=> [3,1]
=> 11 => 00 => 0
[1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0000 => 0
[5]
=> [2,2,1]
=> 001 => 110 => 2
[4,1]
=> [3,2]
=> 10 => 01 => 1
[3,2]
=> [4,1]
=> 01 => 10 => 1
[3,1,1]
=> [2,1,1,1]
=> 0111 => 1000 => 1
[2,2,1]
=> [5]
=> 1 => 0 => 0
[2,1,1,1]
=> [3,1,1]
=> 111 => 000 => 0
[1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 00000 => 0
[6]
=> [2,2,2]
=> 000 => 111 => 3
[5,1]
=> [2,2,1,1]
=> 0011 => 1100 => 2
[4,2]
=> [4,2]
=> 00 => 11 => 2
[4,1,1]
=> [4,1,1]
=> 011 => 100 => 1
[3,3]
=> [3,2,1]
=> 101 => 010 => 1
[3,2,1]
=> [3,3]
=> 11 => 00 => 0
[3,1,1,1]
=> [2,1,1,1,1]
=> 01111 => 10000 => 1
[2,2,2]
=> [6]
=> 0 => 1 => 1
[2,2,1,1]
=> [5,1]
=> 11 => 00 => 0
[2,1,1,1,1]
=> [3,1,1,1]
=> 1111 => 0000 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 111111 => 000000 => 0
[7]
=> [2,2,2,1]
=> 0001 => 1110 => 3
[6,1]
=> [3,2,2]
=> 100 => 011 => 2
[5,2]
=> [4,2,1]
=> 001 => 110 => 2
[5,1,1]
=> [2,2,1,1,1]
=> 00111 => 11000 => 2
[4,3]
=> [4,3]
=> 01 => 10 => 1
[4,2,1]
=> [5,2]
=> 10 => 01 => 1
[4,1,1,1]
=> [4,1,1,1]
=> 0111 => 1000 => 1
[3,3,1]
=> [3,2,1,1]
=> 1011 => 0100 => 1
[3,2,2]
=> [6,1]
=> 01 => 10 => 1
[3,2,1,1]
=> [3,3,1]
=> 111 => 000 => 0
[3,1,1,1,1]
=> [2,1,1,1,1,1]
=> 011111 => 100000 => 1
[2,2,2,1]
=> [7]
=> 1 => 0 => 0
[2,2,1,1,1]
=> [5,1,1]
=> 111 => 000 => 0
[2,1,1,1,1,1]
=> [3,1,1,1,1]
=> 11111 => 00000 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0
[8]
=> [2,2,2,2]
=> 0000 => 1111 => 4
[7,1]
=> [2,2,2,1,1]
=> 00011 => 11100 => 3
[6,2]
=> [4,2,2]
=> 000 => 111 => 3
[6,1,1]
=> [4,2,1,1]
=> 0011 => 1100 => 2
[5,3]
=> [3,2,2,1]
=> 1001 => 0110 => 2
[5,2,1]
=> [3,3,2]
=> 110 => 001 => 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 0000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => ? = 0
[5,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1100000000 => ? = 2
[2,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 0000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => ? = 0
[7,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> 0001111111 => 1110000000 => ? = 3
[5,3,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1]
=> 100111111 => 011000000 => ? = 2
[3,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => 100000000000 => ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? = 0
[6,1,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1100000000 => ? = 2
[5,3,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> 1001111111 => 0110000000 => ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 0000000000 => ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ? = 0
[6,3,1,1,1,1,1,1]
=> [5,2,2,1,1,1,1,1,1]
=> 100111111 => 011000000 => ? = 2
[5,3,1,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1,1,1]
=> 10011111111 => 01100000000 => ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ? = 0
[9,1,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => 111100000000 => ? = 4
[7,5,1,1,1,1]
=> [4,3,2,2,1,1,1,1,1]
=> 010011111 => 101100000 => ? = 3
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111111 => ? => ? = 0
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111111 => ? => ? = 1
[]
=> ?
=> ? => ? => ? = 0
[6,5,4,3,2,1]
=> ?
=> ? => ? => ? = 0
[5,5,4,3,2,1]
=> ?
=> ? => ? => ? = 0
[6,4,4,3,2,1]
=> ?
=> ? => ? => ? = 1
[5,4,4,3,2,1]
=> ?
=> ? => ? => ? = 0
[4,4,4,3,2,1]
=> ?
=> ? => ? => ? = 0
[6,5,3,3,2,1]
=> ?
=> ? => ? => ? = 1
[5,5,3,3,2,1]
=> ?
=> ? => ? => ? = 1
[5,4,3,3,2,1]
=> ?
=> ? => ? => ? = 0
[6,3,3,3,2,1]
=> ?
=> ? => ? => ? = 1
[6,5,4,2,2,1]
=> ?
=> ? => ? => ? = 1
[5,5,4,2,2,1]
=> ?
=> ? => ? => ? = 1
[6,4,4,2,2,1]
=> ?
=> ? => ? => ? = 2
[6,5,2,2,2,1]
=> ?
=> ? => ? => ? = 1
[6,5,4,3,1,1]
=> ?
=> ? => ? => ? = 1
[5,5,4,3,1,1]
=> ?
=> ? => ? => ? = 1
[6,4,4,3,1,1]
=> ?
=> ? => ? => ? = 2
[6,5,3,3,1,1]
=> ?
=> ? => ? => ? = 2
[5,5,3,3,1,1]
=> ?
=> ? => ? => ? = 2
[6,5,4,1,1,1]
=> ?
=> ? => ? => ? = 1
[6,5,4,3,2]
=> ?
=> ? => ? => ? = 1
[5,5,4,3,2]
=> ?
=> ? => ? => ? = 1
[6,4,4,3,2]
=> ?
=> ? => ? => ? = 2
[6,5,3,3,2]
=> ?
=> ? => ? => ? = 2
[5,5,3,3,2]
=> ?
=> ? => ? => ? = 2
[6,5,4,2,2]
=> ?
=> ? => ? => ? = 2
[5,5,4,2,2]
=> ?
=> ? => ? => ? = 2
[6,4,4,2,2]
=> ?
=> ? => ? => ? = 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000992
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 59%●distinct values known / distinct values provided: 29%
Values
[1]
=> [1]
=> [1]
=> []
=> 0
[2]
=> [2]
=> [1,1]
=> [1]
=> 1
[1,1]
=> [1,1]
=> [2]
=> []
=> 0
[3]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1]
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[4]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[3,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,2]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[5]
=> [2,2,1]
=> [3,2]
=> [2]
=> 2
[4,1]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[3,2]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[2,2,1]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[6]
=> [2,2,2]
=> [3,3]
=> [3]
=> 3
[5,1]
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[4,2]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[4,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,3]
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 1
[3,2,1]
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 0
[3,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
[2,2,2]
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[2,2,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1]
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0
[7]
=> [2,2,2,1]
=> [4,3]
=> [3]
=> 3
[6,1]
=> [3,2,2]
=> [3,3,1]
=> [3,1]
=> 2
[5,2]
=> [4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[5,1,1]
=> [2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[4,3]
=> [4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[4,2,1]
=> [5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[3,2,2]
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[3,2,1,1]
=> [3,3,1]
=> [3,2,2]
=> [2,2]
=> 0
[3,1,1,1,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 1
[2,2,2,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[2,2,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1]
=> [3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> []
=> 0
[8]
=> [2,2,2,2]
=> [4,4]
=> [4]
=> 4
[7,1]
=> [2,2,2,1,1]
=> [5,3]
=> [3]
=> 3
[6,2]
=> [4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[6,1,1]
=> [4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[5,3]
=> [3,2,2,1]
=> [4,3,1]
=> [3,1]
=> 2
[5,2,1]
=> [3,3,2]
=> [3,3,2]
=> [3,2]
=> 1
[2,2,2,2,2,2,2]
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,4,4,3]
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> ? = 1
[2,2,2,2,2,2,2,1]
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[6,6,4]
=> [6,6,4]
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> ? = 3
[6,6,2,2]
=> [8,4,4]
=> [3,3,3,3,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> ? = 3
[5,4,4,3]
=> [7,6,3]
=> [3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> ? = 1
[5,4,2,2,2,1]
=> [10,3,3]
=> [3,3,3,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1]
=> ? = 1
[4,4,4,4]
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 2
[4,4,4,2,2]
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 2
[4,4,3,3,2]
=> [9,6,1]
=> [3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 1
[4,4,2,2,2,2]
=> [12,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> ? = 2
[3,3,2,2,2,2,2]
=> [13,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,2,2,2,2,2,2,2]
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[8,8,1]
=> [5,4,4,4]
=> [4,4,4,4,1]
=> [4,4,4,1]
=> ? = 3
[6,5,2,2,2]
=> [10,4,3]
=> [3,3,3,2,1,1,1,1,1,1]
=> [3,3,2,1,1,1,1,1,1]
=> ? = 2
[6,4,4,3]
=> [8,7,2]
=> [3,3,2,2,2,2,2,1]
=> [3,2,2,2,2,2,1]
=> ? = 2
[5,5,2,2,2,1]
=> [8,3,3,3]
=> [4,4,4,1,1,1,1,1]
=> [4,4,1,1,1,1,1]
=> ? = 1
[5,4,4,4]
=> [8,8,1]
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 2
[5,4,3,2,2,1]
=> [7,5,5]
=> [3,3,3,3,3,1,1]
=> [3,3,3,3,1,1]
=> ? = 0
[4,4,4,3,2]
=> [10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,2,1,1,1]
=> ? = 1
[4,4,4,3,1,1]
=> [9,6,1,1]
=> [4,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 1
[4,4,4,2,2,1]
=> [11,6]
=> [2,2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1,1]
=> ? = 1
[4,4,3,3,3]
=> [7,7,2,1]
=> [4,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> ? = 1
[3,2,2,2,2,2,2,2]
=> [16,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[]
=> ?
=> ?
=> ?
=> ? = 0
[6,5,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 0
[5,5,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 0
[6,4,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[5,4,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 0
[4,4,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 0
[6,5,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[5,5,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[5,4,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 0
[6,3,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[6,5,4,2,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[5,5,4,2,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[6,4,4,2,2,1]
=> ?
=> ?
=> ?
=> ? = 2
[6,5,2,2,2,1]
=> ?
=> ?
=> ?
=> ? = 1
[6,5,4,3,1,1]
=> ?
=> ?
=> ?
=> ? = 1
[5,5,4,3,1,1]
=> ?
=> ?
=> ?
=> ? = 1
[6,4,4,3,1,1]
=> ?
=> ?
=> ?
=> ? = 2
[6,5,3,3,1,1]
=> ?
=> ?
=> ?
=> ? = 2
[5,5,3,3,1,1]
=> ?
=> ?
=> ?
=> ? = 2
[6,5,4,1,1,1]
=> ?
=> ?
=> ?
=> ? = 1
[6,5,4,3,2]
=> ?
=> ?
=> ?
=> ? = 1
[5,5,4,3,2]
=> ?
=> ?
=> ?
=> ? = 1
[6,4,4,3,2]
=> ?
=> ?
=> ?
=> ? = 2
[6,5,3,3,2]
=> ?
=> ?
=> ?
=> ? = 2
[5,5,3,3,2]
=> ?
=> ?
=> ?
=> ? = 2
[6,5,4,2,2]
=> ?
=> ?
=> ?
=> ? = 2
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000150
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 57%●distinct values known / distinct values provided: 29%
St000150: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 57%●distinct values known / distinct values provided: 29%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 0
[3]
=> [1,1,1]
=> 1
[2,1]
=> [2,1]
=> 0
[1,1,1]
=> [3]
=> 0
[4]
=> [1,1,1,1]
=> 2
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [4]
=> 0
[5]
=> [1,1,1,1,1]
=> 2
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 1
[2,2,1]
=> [3,2]
=> 0
[2,1,1,1]
=> [4,1]
=> 0
[1,1,1,1,1]
=> [5]
=> 0
[6]
=> [1,1,1,1,1,1]
=> 3
[5,1]
=> [2,1,1,1,1]
=> 2
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> 1
[3,2,1]
=> [3,2,1]
=> 0
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 1
[2,2,1,1]
=> [4,2]
=> 0
[2,1,1,1,1]
=> [5,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> 0
[7]
=> [1,1,1,1,1,1,1]
=> 3
[6,1]
=> [2,1,1,1,1,1]
=> 2
[5,2]
=> [2,2,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> 2
[4,3]
=> [2,2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> 1
[3,2,2]
=> [3,3,1]
=> 1
[3,2,1,1]
=> [4,2,1]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> 1
[2,2,2,1]
=> [4,3]
=> 0
[2,2,1,1,1]
=> [5,2]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> 4
[7,1]
=> [2,1,1,1,1,1,1]
=> 3
[6,2]
=> [2,2,1,1,1,1]
=> 3
[6,1,1]
=> [3,1,1,1,1,1]
=> 2
[5,3]
=> [2,2,2,1,1]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
[3,2,2,1,1,1,1]
=> [7,3,1]
=> ? = 0
[7,4,1]
=> [3,2,2,2,1,1,1]
=> ? = 2
[7,3,2]
=> [3,3,2,1,1,1,1]
=> ? = 3
[6,4,1,1]
=> [4,2,2,2,1,1]
=> ? = 2
[6,2,2,2]
=> [4,4,1,1,1,1]
=> ? = 3
[5,2,1,1,1,1,1]
=> [7,2,1,1,1]
=> ? = 1
[9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> ? = 3
[8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 3
[8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 3
[8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> ? = 3
[7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 2
[7,3,3]
=> [3,3,3,1,1,1,1]
=> ? = 3
[7,3,2,1]
=> [4,3,2,1,1,1,1]
=> ? = 2
[7,2,2,2]
=> [4,4,1,1,1,1,1]
=> ? = 3
[7,2,1,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
[6,4,2,1]
=> [4,3,2,2,1,1]
=> ? = 2
[6,3,3,1]
=> [4,3,3,1,1,1]
=> ? = 2
[6,3,2,2]
=> [4,4,2,1,1,1]
=> ? = 2
[6,1,1,1,1,1,1,1]
=> [8,1,1,1,1,1]
=> ? = 2
[5,5,2,1]
=> [4,3,2,2,2]
=> ? = 1
[5,3,2,1,1,1]
=> [6,3,2,1,1]
=> ? = 1
[5,3,1,1,1,1,1]
=> [7,2,2,1,1]
=> ? = 2
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> ? = 1
[5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> ? = 1
[5,2,1,1,1,1,1,1]
=> [8,2,1,1,1]
=> ? = 1
[3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> ? = 0
[3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [11,1,1]
=> ? = 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> ? = 0
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 5
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> ? = 4
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> ? = 5
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 4
[9,4,1]
=> [3,2,2,2,1,1,1,1,1]
=> ? = 3
[9,2,2,1]
=> [4,3,1,1,1,1,1,1,1]
=> ? = 3
[8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 4
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 4
[8,3,3]
=> [3,3,3,1,1,1,1,1]
=> ? = 3
[8,1,1,1,1,1,1]
=> [7,1,1,1,1,1,1,1]
=> ? = 3
[7,6,1]
=> [3,2,2,2,2,2,1]
=> ? = 2
[7,5,2]
=> [3,3,2,2,2,1,1]
=> ? = 3
[7,4,3]
=> [3,3,3,2,1,1,1]
=> ? = 2
[6,6,1,1]
=> [4,2,2,2,2,2]
=> ? = 2
[6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 3
[6,1,1,1,1,1,1,1,1]
=> [9,1,1,1,1,1]
=> ? = 2
[5,4,4,1]
=> [4,3,3,3,1]
=> ? = 1
[5,4,2,1,1,1]
=> [6,3,2,2,1]
=> ? = 1
[5,3,2,2,2]
=> [5,5,2,1,1]
=> ? = 2
[5,3,2,2,1,1]
=> [6,4,2,1,1]
=> ? = 1
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000142
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 38%●distinct values known / distinct values provided: 29%
St000142: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 38%●distinct values known / distinct values provided: 29%
Values
[1]
=> [1]
=> 0
[2]
=> [2]
=> 1
[1,1]
=> [1,1]
=> 0
[3]
=> [2,1]
=> 1
[2,1]
=> [3]
=> 0
[1,1,1]
=> [1,1,1]
=> 0
[4]
=> [2,2]
=> 2
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [4]
=> 1
[2,1,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [1,1,1,1]
=> 0
[5]
=> [2,2,1]
=> 2
[4,1]
=> [3,2]
=> 1
[3,2]
=> [4,1]
=> 1
[3,1,1]
=> [2,1,1,1]
=> 1
[2,2,1]
=> [5]
=> 0
[2,1,1,1]
=> [3,1,1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[6]
=> [2,2,2]
=> 3
[5,1]
=> [2,2,1,1]
=> 2
[4,2]
=> [4,2]
=> 2
[4,1,1]
=> [4,1,1]
=> 1
[3,3]
=> [3,2,1]
=> 1
[3,2,1]
=> [3,3]
=> 0
[3,1,1,1]
=> [2,1,1,1,1]
=> 1
[2,2,2]
=> [6]
=> 1
[2,2,1,1]
=> [5,1]
=> 0
[2,1,1,1,1]
=> [3,1,1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7]
=> [2,2,2,1]
=> 3
[6,1]
=> [3,2,2]
=> 2
[5,2]
=> [4,2,1]
=> 2
[5,1,1]
=> [2,2,1,1,1]
=> 2
[4,3]
=> [4,3]
=> 1
[4,2,1]
=> [5,2]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,1,1]
=> 1
[3,2,2]
=> [6,1]
=> 1
[3,2,1,1]
=> [3,3,1]
=> 0
[3,1,1,1,1]
=> [2,1,1,1,1,1]
=> 1
[2,2,2,1]
=> [7]
=> 0
[2,2,1,1,1]
=> [5,1,1]
=> 0
[2,1,1,1,1,1]
=> [3,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
[8]
=> [2,2,2,2]
=> 4
[7,1]
=> [2,2,2,1,1]
=> 3
[6,2]
=> [4,2,2]
=> 3
[6,1,1]
=> [4,2,1,1]
=> 2
[5,3]
=> [3,2,2,1]
=> 2
[5,2,1]
=> [3,3,2]
=> 1
[9,1,1]
=> [2,2,2,2,1,1,1]
=> ? = 4
[8,1,1,1]
=> [4,2,2,1,1,1]
=> ? = 3
[7,2,2]
=> [6,2,2,1]
=> ? = 3
[6,3,2]
=> [6,3,2]
=> ? = 2
[6,2,1,1,1]
=> [6,2,1,1,1]
=> ? = 2
[6,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
[5,3,1,1,1]
=> [3,2,2,1,1,1,1]
=> ? = 2
[5,2,2,2]
=> [8,2,1]
=> ? = 2
[5,2,2,1,1]
=> [6,3,1,1]
=> ? = 1
[5,2,1,1,1,1]
=> [4,3,1,1,1,1]
=> ? = 1
[5,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 2
[4,4,1,1,1]
=> [7,2,1,1]
=> ? = 1
[4,2,2,2,1]
=> [9,2]
=> ? = 1
[4,2,2,1,1,1]
=> [8,1,1,1]
=> ? = 1
[4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> ? = 1
[3,3,3,1,1]
=> [3,3,2,1,1,1]
=> ? = 1
[3,3,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> ? = 1
[3,2,2,2,1,1]
=> [7,3,1]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [7,1,1,1,1]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 0
[11,1]
=> [2,2,2,2,2,1,1]
=> ? = 5
[10,2]
=> [4,2,2,2,2]
=> ? = 5
[10,1,1]
=> [4,2,2,2,1,1]
=> ? = 4
[9,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> ? = 4
[8,1,1,1,1]
=> [4,2,2,1,1,1,1]
=> ? = 3
[7,3,1,1]
=> [3,2,2,2,1,1,1]
=> ? = 3
[7,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 3
[6,4,1,1]
=> [6,4,1,1]
=> ? = 2
[6,2,2,1,1]
=> [8,2,1,1]
=> ? = 2
[6,2,1,1,1,1]
=> [6,2,1,1,1,1]
=> ? = 2
[6,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1]
=> ? = 2
[5,5,2]
=> [6,3,2,1]
=> ? = 2
[5,5,1,1]
=> [4,3,2,1,1,1]
=> ? = 2
[5,3,2,2]
=> [7,2,2,1]
=> ? = 2
[5,3,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> ? = 2
[5,2,2,2,1]
=> [7,3,2]
=> ? = 1
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> ? = 1
[5,2,1,1,1,1,1]
=> [4,3,1,1,1,1,1]
=> ? = 1
[4,4,3,1]
=> [6,5,1]
=> ? = 1
[4,4,2,2]
=> [8,4]
=> ? = 2
[4,4,2,1,1]
=> [8,3,1]
=> ? = 1
[4,3,1,1,1,1,1]
=> [5,2,1,1,1,1,1]
=> ? = 1
[4,2,2,1,1,1,1]
=> [8,1,1,1,1]
=> ? = 1
[4,2,1,1,1,1,1,1]
=> [6,1,1,1,1,1,1]
=> ? = 1
[4,1,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> ? = 1
[3,3,3,1,1,1]
=> [3,3,2,1,1,1,1]
=> ? = 1
[3,3,2,2,2]
=> [9,2,1]
=> ? = 1
[3,3,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> ? = 1
[3,2,2,1,1,1,1,1]
=> [5,3,1,1,1,1]
=> ? = 0
Description
The number of even parts of a partition.
Matching statistic: St001487
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 10%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 2 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? = 2 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ? = 3 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? = 2 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ? = 3 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ? = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ? = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 2 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ? = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ? = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ? = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ? = 4 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ? = 3 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ? = 3 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ? = 2 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ? = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ? = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? = 2 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? = 2 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> ? = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> ? = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,1],[]]
=> ? = 1 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1]]
=> ? = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5,1],[]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[5,5,5,5,1],[]]
=> ? = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> ? = 4 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> ? = 3 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> ? = 3 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> ? = 3 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> ? = 2 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> ? = 2 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> ? = 2 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 2 = 1 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 2 = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
Description
The number of inner corners of a skew partition.
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