- FindStat

Identifier

Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000293: Binary words ⟶ ℤ

Values

[[1],[]] => [1] => 10 => 01 => 0

[[2],[]] => [2] => 100 => 011 => 0

[[1,1],[]] => [1,1] => 110 => 001 => 0

[[2,1],[1]] => [2,1] => 1010 => 0101 => 1

[[3],[]] => [3] => 1000 => 0111 => 0

[[2,1],[]] => [2,1] => 1010 => 0101 => 1

[[3,1],[1]] => [3,1] => 10010 => 01101 => 2

[[2,2],[1]] => [2,2] => 1100 => 0011 => 0

[[3,2],[2]] => [3,2] => 10100 => 01011 => 1

[[1,1,1],[]] => [1,1,1] => 1110 => 0001 => 0

[[2,2,1],[1,1]] => [2,2,1] => 11010 => 00101 => 1

[[2,1,1],[1]] => [2,1,1] => 10110 => 01001 => 2

[[3,2,1],[2,1]] => [3,2,1] => 101010 => 010101 => 3

[[4],[]] => [4] => 10000 => 01111 => 0

[[3,1],[]] => [3,1] => 10010 => 01101 => 2

[[4,1],[1]] => [4,1] => 100010 => 011101 => 3

[[2,2],[]] => [2,2] => 1100 => 0011 => 0

[[3,2],[1]] => [3,2] => 10100 => 01011 => 1

[[4,2],[2]] => [4,2] => 100100 => 011011 => 2

[[2,1,1],[]] => [2,1,1] => 10110 => 01001 => 2

[[3,2,1],[1,1]] => [3,2,1] => 101010 => 010101 => 3

[[3,1,1],[1]] => [3,1,1] => 100110 => 011001 => 4

[[4,2,1],[2,1]] => [4,2,1] => 1001010 => 0110101 => 5

[[3,3],[2]] => [3,3] => 11000 => 00111 => 0

[[4,3],[3]] => [4,3] => 101000 => 010111 => 1

[[2,2,1],[1]] => [2,2,1] => 11010 => 00101 => 1

[[3,3,1],[2,1]] => [3,3,1] => 110010 => 001101 => 2

[[3,2,1],[2]] => [3,2,1] => 101010 => 010101 => 3

[[4,3,1],[3,1]] => [4,3,1] => 1010010 => 0101101 => 4

[[2,2,2],[1,1]] => [2,2,2] => 11100 => 00011 => 0

[[3,3,2],[2,2]] => [3,3,2] => 110100 => 001011 => 1

[[3,2,2],[2,1]] => [3,2,2] => 101100 => 010011 => 2

[[4,3,2],[3,2]] => [4,3,2] => 1010100 => 0101011 => 3

[[1,1,1,1],[]] => [1,1,1,1] => 11110 => 00001 => 0

[[2,2,2,1],[1,1,1]] => [2,2,2,1] => 111010 => 000101 => 1

[[2,2,1,1],[1,1]] => [2,2,1,1] => 110110 => 001001 => 2

[[3,3,2,1],[2,2,1]] => [3,3,2,1] => 1101010 => 0010101 => 3

[[2,1,1,1],[1]] => [2,1,1,1] => 101110 => 010001 => 3

[[3,2,2,1],[2,1,1]] => [3,2,2,1] => 1011010 => 0100101 => 4

[[3,2,1,1],[2,1]] => [3,2,1,1] => 1010110 => 0101001 => 5

[[4,3,2,1],[3,2,1]] => [4,3,2,1] => 10101010 => 01010101 => 6

[[5],[]] => [5] => 100000 => 011111 => 0

[[4,1],[]] => [4,1] => 100010 => 011101 => 3

[[5,1],[1]] => [5,1] => 1000010 => 0111101 => 4

[[3,2],[]] => [3,2] => 10100 => 01011 => 1

[[4,2],[1]] => [4,2] => 100100 => 011011 => 2

[[5,2],[2]] => [5,2] => 1000100 => 0111011 => 3

[[3,1,1],[]] => [3,1,1] => 100110 => 011001 => 4

[[4,2,1],[1,1]] => [4,2,1] => 1001010 => 0110101 => 5

[[4,1,1],[1]] => [4,1,1] => 1000110 => 0111001 => 6

[[5,2,1],[2,1]] => [5,2,1] => 10001010 => 01110101 => 7

[[3,3],[1]] => [3,3] => 11000 => 00111 => 0

[[4,3],[2]] => [4,3] => 101000 => 010111 => 1

[[5,3],[3]] => [5,3] => 1001000 => 0110111 => 2

[[2,2,1],[]] => [2,2,1] => 11010 => 00101 => 1

[[3,3,1],[1,1]] => [3,3,1] => 110010 => 001101 => 2

[[3,2,1],[1]] => [3,2,1] => 101010 => 010101 => 3

[[4,3,1],[2,1]] => [4,3,1] => 1010010 => 0101101 => 4

[[4,2,1],[2]] => [4,2,1] => 1001010 => 0110101 => 5

[[5,3,1],[3,1]] => [5,3,1] => 10010010 => 01101101 => 6

[[3,2,2],[1,1]] => [3,2,2] => 101100 => 010011 => 2

[[4,3,2],[2,2]] => [4,3,2] => 1010100 => 0101011 => 3

[[4,2,2],[2,1]] => [4,2,2] => 1001100 => 0110011 => 4

[[5,3,2],[3,2]] => [5,3,2] => 10010100 => 01101011 => 5

[[2,1,1,1],[]] => [2,1,1,1] => 101110 => 010001 => 3

[[3,2,2,1],[1,1,1]] => [3,2,2,1] => 1011010 => 0100101 => 4

[[3,2,1,1],[1,1]] => [3,2,1,1] => 1010110 => 0101001 => 5

[[4,3,2,1],[2,2,1]] => [4,3,2,1] => 10101010 => 01010101 => 6

[[3,1,1,1],[1]] => [3,1,1,1] => 1001110 => 0110001 => 6

[[4,2,2,1],[2,1,1]] => [4,2,2,1] => 10011010 => 01100101 => 7

[[4,2,1,1],[2,1]] => [4,2,1,1] => 10010110 => 01101001 => 8

[[5,3,2,1],[3,2,1]] => [5,3,2,1] => 100101010 => 011010101 => 9

[[4,4],[3]] => [4,4] => 110000 => 001111 => 0

[[5,4],[4]] => [5,4] => 1010000 => 0101111 => 1

[[3,3,1],[2]] => [3,3,1] => 110010 => 001101 => 2

[[4,4,1],[3,1]] => [4,4,1] => 1100010 => 0011101 => 3

[[4,3,1],[3]] => [4,3,1] => 1010010 => 0101101 => 4

[[5,4,1],[4,1]] => [5,4,1] => 10100010 => 01011101 => 5

[[2,2,2],[1]] => [2,2,2] => 11100 => 00011 => 0

[[3,3,2],[2,1]] => [3,3,2] => 110100 => 001011 => 1

[[4,4,2],[3,2]] => [4,4,2] => 1100100 => 0011011 => 2

[[3,2,2],[2]] => [3,2,2] => 101100 => 010011 => 2

[[4,3,2],[3,1]] => [4,3,2] => 1010100 => 0101011 => 3

[[5,4,2],[4,2]] => [5,4,2] => 10100100 => 01011011 => 4

[[2,2,1,1],[1]] => [2,2,1,1] => 110110 => 001001 => 2

[[3,3,2,1],[2,1,1]] => [3,3,2,1] => 1101010 => 0010101 => 3

[[3,3,1,1],[2,1]] => [3,3,1,1] => 1100110 => 0011001 => 4

[[4,4,2,1],[3,2,1]] => [4,4,2,1] => 11001010 => 00110101 => 5

[[3,2,1,1],[2]] => [3,2,1,1] => 1010110 => 0101001 => 5

[[4,3,2,1],[3,1,1]] => [4,3,2,1] => 10101010 => 01010101 => 6

[[4,3,1,1],[3,1]] => [4,3,1,1] => 10100110 => 01011001 => 7

[[5,4,2,1],[4,2,1]] => [5,4,2,1] => 101001010 => 010110101 => 8

[[3,3,3],[2,2]] => [3,3,3] => 111000 => 000111 => 0

[[4,4,3],[3,3]] => [4,4,3] => 1101000 => 0010111 => 1

[[4,3,3],[3,2]] => [4,3,3] => 1011000 => 0100111 => 2

[[5,4,3],[4,3]] => [5,4,3] => 10101000 => 01010111 => 3

[[2,2,2,1],[1,1]] => [2,2,2,1] => 111010 => 000101 => 1

[[3,3,3,1],[2,2,1]] => [3,3,3,1] => 1110010 => 0001101 => 2

[[3,3,2,1],[2,2]] => [3,3,2,1] => 1101010 => 0010101 => 3

[[4,4,3,1],[3,3,1]] => [4,4,3,1] => 11010010 => 00101101 => 4

[[3,2,2,1],[2,1]] => [3,2,2,1] => 1011010 => 0100101 => 4

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$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 3 + 3\ q + 2\ q^{2} + q^{3}$

$F_{4} = 5 + 5\ q + 6\ q^{2} + 6\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6}$

$F_{5} = 7 + 9\ q + 12\ q^{2} + 14\ q^{3} + 13\ q^{4} + 11\ q^{5} + 9\ q^{6} + 6\ q^{7} + 3\ q^{8} + 2\ q^{9} + q^{10}$

Description

The number of inversions of a binary word.

Map

outer shape

Description

The outer shape of the skew partition.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.