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Your data matches 128 different statistics following compositions of up to 3 maps.
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Matching statistic: St000897
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 1
[3,2,1]
=> 1
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 2
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000159
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 2 = 1 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 2 = 1 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 2 = 1 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 3 = 2 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 2 = 1 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 3 = 2 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 3 = 2 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 3 = 2 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 2 = 1 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 3 = 2 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 2 = 1 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 3 = 2 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 2 = 1 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 2 = 1 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 2 = 1 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 3 = 2 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 3 = 2 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 3 = 2 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 3 = 2 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 3 = 2 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 3 = 2 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 3 = 2 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 3 = 2 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 2 = 1 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 3 = 2 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 3 = 2 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 3 = 2 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 3 = 2 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 3 = 2 + 1
[11]
=> 100000000000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[10,1]
=> 100000000010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[9,1,1]
=> 100000000110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[8,3]
=> 1000001000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[8,2,1]
=> 10000001010 => [1,7,2,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[8,1,1,1]
=> 100000001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,3,1]
=> 1000010010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,2,2]
=> 1000001100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,2,1,1]
=> 10000010110 => [1,6,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[5,2,1,1,1,1]
=> 10001011110 => [1,4,2,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => [1,3,2,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => [1,2,2,1,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => [1,1,1,2,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
[12]
=> 1000000000000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[11,1]
=> 1000000000010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[10,2]
=> 100000000100 => [1,9,3] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[10,1,1]
=> 1000000000110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[9,3]
=> 10000001000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[9,2,1]
=> 100000001010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[9,1,1,1]
=> 1000000001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[8,3,1]
=> 10000010010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[8,2,2]
=> 10000001100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[8,2,1,1]
=> 100000010110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[7,4,1]
=> 1000100010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[7,3,2]
=> 1000010100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[7,3,1,1]
=> 10000100110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[7,2,2,1]
=> 10000011010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[7,2,1,1,1]
=> 100000101110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[7,1,1,1,1,1]
=> 1000000111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[6,4,1,1]
=> 1001000110 => [1,3,4,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[6,3,2,1]
=> 1000101010 => [1,4,2,2,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[6,3,1,1,1]
=> 10001001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
[6,2,2,2]
=> 1000011100 => [1,5,1,1,3] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3} + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St001568
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1}
[2,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1}
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,2}
[3,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,2}
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,2,2}
[4,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2,2}
[3,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2}
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,2,2}
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,2,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,1,1,2,2}
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1}
[5,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1}
[4,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1}
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1}
[3,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1}
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1}
[2,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1}
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[6,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[5,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[4,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[3,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[3,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[7,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[6,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[5,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[4,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[4,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[4,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[3,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[8,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[7,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[6,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[6,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[5,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[5,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2}
[5,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,4,1]
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[4,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[4,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[3,3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[3,3,1,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> [2,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 2
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[7,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[6,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[5,4,1]
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[5,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2
[5,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001124
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1
[2]
=> 100 => [1,3] => [3,1]
=> 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 2
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 2
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 2
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 2
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 2
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 2
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 2
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 2
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 2
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 2
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 2
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 2
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 2
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 2
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 2
[9]
=> 1000000000 => [1,10] => [10,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[8,1]
=> 1000000010 => [1,8,2] => [8,2,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[7,1,1]
=> 1000000110 => [1,7,1,2] => [7,2,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [6,2,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [5,2,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [4,2,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[3,1,1,1,1,1,1]
=> 1001111110 => [1,3,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1]
=> 1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2}
[10]
=> 10000000000 => [1,11] => [11,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[9,1]
=> 10000000010 => [1,9,2] => [9,2,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[8,2]
=> 1000000100 => [1,7,3] => [7,3,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[8,1,1]
=> 10000000110 => [1,8,1,2] => [8,2,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[7,2,1]
=> 1000001010 => [1,6,2,2] => [6,2,2,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[7,1,1,1]
=> 10000001110 => [1,7,1,1,2] => [7,2,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[6,1,1,1,1]
=> 10000011110 => [1,6,1,1,1,2] => [6,2,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,2,1,1,1]
=> 1000101110 => [1,4,2,1,1,2] => [4,2,2,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,1,1,1,1]
=> 10000111110 => [1,5,1,1,1,1,2] => [5,2,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,2,1,1,1,1]
=> 1001011110 => [1,3,2,1,1,1,2] => [3,2,2,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,1,1,1,1,1]
=> 10001111110 => [1,4,1,1,1,1,1,2] => [4,2,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,1,1,1,1]
=> 1010111110 => [1,2,2,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1,1,1]
=> 10011111110 => [1,3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1,1,1]
=> 1101111110 => [1,1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => [1,2,1,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => [1,1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[11]
=> 100000000000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[10,1]
=> 100000000010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[9,2]
=> 10000000100 => [1,8,3] => [8,3,1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[9,1,1]
=> 100000000110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[8,3]
=> 1000001000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[8,2,1]
=> 10000001010 => [1,7,2,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[8,1,1,1]
=> 100000001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[7,3,1]
=> 1000010010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[7,2,2]
=> 1000001100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[7,2,1,1]
=> 10000010110 => [1,6,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[7,1,1,1,1]
=> 100000011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,2,1,1,1,1]
=> 10001011110 => [1,4,2,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,3,1,1,1,1]
=> 1010011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,2,1,1,1]
=> 1001101110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,2,1,1,1,1,1]
=> 10010111110 => [1,3,2,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The multiplicity 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 $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000318
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 3 = 1 + 2
[2]
=> 100 => [1,3] => [3,1]
=> 3 = 1 + 2
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 3 = 1 + 2
[3]
=> 1000 => [1,4] => [4,1]
=> 3 = 1 + 2
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 3 = 1 + 2
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 1 + 2
[4]
=> 10000 => [1,5] => [5,1]
=> 3 = 1 + 2
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 4 = 2 + 2
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 3 = 1 + 2
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 3 = 1 + 2
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 3 = 1 + 2
[5]
=> 100000 => [1,6] => [6,1]
=> 3 = 1 + 2
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 4 = 2 + 2
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 4 = 2 + 2
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 4 = 2 + 2
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 3 = 1 + 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 1 + 2
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 3 = 1 + 2
[6]
=> 1000000 => [1,7] => [7,1]
=> 3 = 1 + 2
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 4 = 2 + 2
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 3 = 1 + 2
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 4 = 2 + 2
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 3 = 1 + 2
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 3 = 1 + 2
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 4 = 2 + 2
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 1 + 2
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 1 + 2
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 3 = 1 + 2
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 3 = 1 + 2
[7]
=> 10000000 => [1,8] => [8,1]
=> 3 = 1 + 2
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 4 = 2 + 2
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 4 = 2 + 2
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 4 = 2 + 2
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 4 = 2 + 2
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 4 = 2 + 2
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 4 = 2 + 2
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 4 = 2 + 2
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 4 = 2 + 2
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 3 = 1 + 2
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 2 + 2
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 1 + 2
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 3 = 1 + 2
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 3 = 1 + 2
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 3 = 1 + 2
[8]
=> 100000000 => [1,9] => [9,1]
=> 3 = 1 + 2
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 4 = 2 + 2
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 4 = 2 + 2
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 4 = 2 + 2
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 4 = 2 + 2
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 4 = 2 + 2
[9]
=> 1000000000 => [1,10] => [10,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[8,1]
=> 1000000010 => [1,8,2] => [8,2,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[7,1,1]
=> 1000000110 => [1,7,1,2] => [7,2,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [6,2,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [5,2,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [4,2,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[3,1,1,1,1,1,1]
=> 1001111110 => [1,3,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[2,1,1,1,1,1,1,1]
=> 1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,2,2,2,2,2,2} + 2
[10]
=> 10000000000 => [1,11] => [11,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[9,1]
=> 10000000010 => [1,9,2] => [9,2,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[8,2]
=> 1000000100 => [1,7,3] => [7,3,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[8,1,1]
=> 10000000110 => [1,8,1,2] => [8,2,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[7,2,1]
=> 1000001010 => [1,6,2,2] => [6,2,2,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[7,1,1,1]
=> 10000001110 => [1,7,1,1,2] => [7,2,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[6,1,1,1,1]
=> 10000011110 => [1,6,1,1,1,2] => [6,2,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[5,2,1,1,1]
=> 1000101110 => [1,4,2,1,1,2] => [4,2,2,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[5,1,1,1,1,1]
=> 10000111110 => [1,5,1,1,1,1,2] => [5,2,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[4,2,1,1,1,1]
=> 1001011110 => [1,3,2,1,1,1,2] => [3,2,2,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[4,1,1,1,1,1,1]
=> 10001111110 => [1,4,1,1,1,1,1,2] => [4,2,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[3,2,1,1,1,1,1]
=> 1010111110 => [1,2,2,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[3,1,1,1,1,1,1,1]
=> 10011111110 => [1,3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[2,2,1,1,1,1,1,1]
=> 1101111110 => [1,1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => [1,2,1,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => [1,1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 2
[11]
=> 100000000000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[10,1]
=> 100000000010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[9,2]
=> 10000000100 => [1,8,3] => [8,3,1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[9,1,1]
=> 100000000110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[8,3]
=> 1000001000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[8,2,1]
=> 10000001010 => [1,7,2,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[8,1,1,1]
=> 100000001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[7,3,1]
=> 1000010010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[7,2,2]
=> 1000001100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[7,2,1,1]
=> 10000010110 => [1,6,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[7,1,1,1,1]
=> 100000011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[5,2,1,1,1,1]
=> 10001011110 => [1,4,2,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[4,3,1,1,1,1]
=> 1010011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[4,2,2,1,1,1]
=> 1001101110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[4,2,1,1,1,1,1]
=> 10010111110 => [1,3,2,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000903
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 1
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 1
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 1
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 2
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 2
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 2
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 1
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [5,4,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [5,3,3] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [5,3,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [4,3,3,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of different parts of an integer composition.
Matching statistic: St001114
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,6,2,3,4,5] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? ∊ {1,2} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,5,2,3,4] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,5,6,2,3,4] => 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? ∊ {1,2} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => ? ∊ {1,2,2,2} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? ∊ {1,2,2,2} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [1,6,2,3,5,4] => 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [1,6,2,3,5,4] => 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,4,5,2,3] => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => 0 = 1 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,5,2,3,4,6] => 0 = 1 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,4,6,2,3,5] => 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,6,7,2,3,4,5] => ? ∊ {1,2,2,2} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,2,3,4,5,6,7] => ? ∊ {1,2,2,2} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [1,7,2,3,4,6,5] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [1,6,2,5,3,4] => 0 = 1 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [1,6,2,5,3,4] => 0 = 1 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [1,6,2,3,4,5] => 0 = 1 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 1 = 2 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,5,2,3,6,4] => 1 = 2 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => 1 = 2 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 0 = 1 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => [1,4,5,6,2,3] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [1,6,2,3,4,5,7] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [1,5,7,2,3,4,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,7,8,2,3,4,5,6] => ? ∊ {1,2,2,2,2,2,2,2} - 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]
=> [1,9,8,7,6,5,4,3,2] => [1,9,2,3,4,5,6,7,8] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,5,7,4,3,2,1,8] => [1,8,2,3,4,5,7,6] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,6,4,3,2,1,8] => [1,8,2,3,4,5,7,6] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [1,6,2,3,4,7,5] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [1,5,6,7,2,3,4] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [1,7,2,3,4,5,6,8] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [1,4,7,2,3,5,6] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => [1,6,8,2,3,4,5,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,8,9,2,3,4,5,6,7] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,10,2,3,4,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [1,11,2,3,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [1,9,2,3,4,5,6,8,7] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,7,5,4,3,2,1,9] => [1,9,2,3,4,5,6,8,7] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,5,4,7,3,2,1,8] => [1,8,2,3,4,7,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,6,5,3,2,1,8] => [1,8,2,3,4,7,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => [1,7,2,3,6,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [1,7,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [1,7,2,6,3,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [1,6,2,3,7,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [1,5,6,2,3,4,7] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [1,7,2,3,4,5,8,6] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [1,5,2,3,4,7,6] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [1,4,6,7,2,3,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,5,4,3,2] => [1,6,7,8,2,3,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,8,2,3,4,5,6,7,9] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,7,6,4,3,2] => [1,5,8,2,3,4,6,7] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
Description
The number of odd descents of a permutation.
Matching statistic: St001487
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ?
=> ? = 1
[2]
=> []
=> []
=> ?
=> ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[3]
=> []
=> []
=> ?
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[4]
=> []
=> []
=> ?
=> ? = 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[5]
=> []
=> []
=> ?
=> ? ∊ {2,2}
[4,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {2,2}
[6]
=> []
=> []
=> ?
=> ? ∊ {1,1,1}
[5,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {1,1,1}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {1,1,1}
[7]
=> []
=> []
=> ?
=> ? ∊ {2,2,2,2,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {2,2,2,2,2}
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {2,2,2,2,2}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {2,2,2,2,2}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? ∊ {2,2,2,2,2}
[8]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[7,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1]
=> [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],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2}
[9]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[8,1]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1,1]
=> [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],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1]
=> [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],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> [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],[]]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2}
[10]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[6,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of inner corners of a skew partition.
Matching statistic: St000872
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,4,3,2] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [1,6,5,4,3,2] => 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => ? ∊ {1,1} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 2 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,4,3,1] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => ? ∊ {1,1} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => [1,8,7,6,5,4,3,2] => ? ∊ {2,2,2,2} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => ? ∊ {2,2,2,2} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [1,6,5,4,3,2] => 0 = 1 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,1,6,5,3,2] => 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,5,1,4,2] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,5,4,2] => 0 = 1 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 1 = 2 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [2,1,6,5,4,3] => 0 = 1 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [2,6,1,5,4,3] => 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => ? ∊ {2,2,2,2} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,8,7,6,5,4,3] => ? ∊ {2,2,2,2} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [1,9,8,7,6,5,4,3,2] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => [1,7,6,5,4,3,2] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => [5,1,7,6,4,3,2] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [1,6,5,4,3,2] => 0 = 1 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => [4,6,1,5,3,2] => 1 = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => 0 = 1 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [1,6,5,4,3,2] => 0 = 1 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 0 = 1 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [2,1,6,5,4,3] => 0 = 1 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 0 = 1 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 0 = 1 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 1 = 2 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [2,6,5,1,4,3] => 1 = 2 - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => [2,1,7,6,5,4,3] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => [2,7,1,6,5,4,3] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,8,1,7,6,5,4,3] => ? ∊ {1,2,2,2,2,2,2,2} - 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]
=> [2,1,3,4,5,6,7,8,9] => [2,1,9,8,7,6,5,4,3] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,10,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [8,1,9,7,6,5,4,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,7,2,3,4,5,8,6] => [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [6,1,8,7,5,4,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => [5,7,1,6,4,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [2,1,7,6,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => [2,7,6,1,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [2,1,8,7,6,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [2,7,1,6,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,8,1,7,6,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [2,9,1,8,7,6,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2} - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,11,10,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => [9,1,10,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,9,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [7,1,9,8,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,2,7,3,4,5,8,6] => [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => [6,8,1,7,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [5,1,8,7,6,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => [5,1,7,6,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,7,6] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,6,1,2,3,7,5] => [4,7,1,6,5,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => [2,1,7,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,6,1,7,3,4,5] => [2,7,1,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [2,1,8,7,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [2,1,7,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [2,7,6,1,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [2,8,7,1,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [2,1,9,8,7,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [2,7,1,6,5,4,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} - 1
Description
The number of very big descents of a permutation.
A very big descent of a permutation $\pi$ is an index $i$ such that $\pi_i - \pi_{i+1} > 2$.
For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General $r$-descents were for example be studied in [1, Section 2].
Matching statistic: St001165
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {1,2,2,2} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,2,2,2} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2} + 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]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
The following 118 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000905The number of different multiplicities of parts of an integer composition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St001469The holeyness of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000741The Colin de Verdière graph invariant. St001665The number of pure excedances of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000834The number of right outer peaks of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001394The genus of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000352The Elizalde-Pak rank of a permutation. St000007The number of saliances of the permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000092The number of outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001728The number of invisible descents of a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001344The neighbouring number of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000061The number of nodes on the left branch of a binary tree. St000099The number of valleys of a permutation, including the boundary. St000252The number of nodes of degree 3 of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000664The number of right ropes of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001715The number of non-records in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000454The largest eigenvalue of a graph if it is integral. St000058The order of a permutation. St000871The number of very big ascents of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001490The number of connected components of a skew partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St000326The position of the first one in a binary word after appending a 1 at the end. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000629The defect of a binary word. St001330The hat guessing number 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. St001896The number of right descents of a signed permutations. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition.
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