- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

[[1]] => [1] => 10 => 1

[[1],[1]] => [1,1] => 110 => 2

[[2]] => [2] => 100 => 1

[[1,1]] => [2] => 100 => 1

[[1],[1],[1]] => [1,1,1] => 1110 => 3

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

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

[[3]] => [3] => 1000 => 1

[[2,1]] => [3] => 1000 => 1

[[1,1,1]] => [3] => 1000 => 1

[[1],[1],[1],[1]] => [1,1,1,1] => 11110 => 4

[[2],[1],[1]] => [2,1,1] => 10110 => 3

[[2],[2]] => [2,2] => 1100 => 2

[[1,1],[1],[1]] => [2,1,1] => 10110 => 3

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

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

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

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

[[4]] => [4] => 10000 => 1

[[3,1]] => [4] => 10000 => 1

[[2,2]] => [4] => 10000 => 1

[[2,1,1]] => [4] => 10000 => 1

[[1,1,1,1]] => [4] => 10000 => 1

[[1],[1],[1],[1],[1]] => [1,1,1,1,1] => 111110 => 5

[[2],[1],[1],[1]] => [2,1,1,1] => 101110 => 4

[[2],[2],[1]] => [2,2,1] => 11010 => 3

[[1,1],[1],[1],[1]] => [2,1,1,1] => 101110 => 4

[[1,1],[1,1],[1]] => [2,2,1] => 11010 => 3

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

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

[[2,1],[1],[1]] => [3,1,1] => 100110 => 3

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

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

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

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

[[4],[1]] => [4,1] => 100010 => 2

[[3,1],[1]] => [4,1] => 100010 => 2

[[2,2],[1]] => [4,1] => 100010 => 2

[[2,1,1],[1]] => [4,1] => 100010 => 2

[[1,1,1,1],[1]] => [4,1] => 100010 => 2

[[5]] => [5] => 100000 => 1

[[4,1]] => [5] => 100000 => 1

[[3,2]] => [5] => 100000 => 1

[[3,1,1]] => [5] => 100000 => 1

[[2,2,1]] => [5] => 100000 => 1

[[2,1,1,1]] => [5] => 100000 => 1

[[1,1,1,1,1]] => [5] => 100000 => 1

[[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => 1111110 => 6

[[2],[1],[1],[1],[1]] => [2,1,1,1,1] => 1011110 => 5

[[2],[2],[1],[1]] => [2,2,1,1] => 110110 => 4

[[2],[2],[2]] => [2,2,2] => 11100 => 3

[[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => 1011110 => 5

[[1,1],[1,1],[1],[1]] => [2,2,1,1] => 110110 => 4

[[1,1],[1,1],[1,1]] => [2,2,2] => 11100 => 3

[[3],[1],[1],[1]] => [3,1,1,1] => 1001110 => 4

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

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

[[2,1],[1],[1],[1]] => [3,1,1,1] => 1001110 => 4

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

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

[[2,1],[2,1]] => [3,3] => 11000 => 2

[[1,1,1],[1],[1],[1]] => [3,1,1,1] => 1001110 => 4

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

[[1,1,1],[1,1,1]] => [3,3] => 11000 => 2

[[4],[1],[1]] => [4,1,1] => 1000110 => 3

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

[[3,1],[1],[1]] => [4,1,1] => 1000110 => 3

[[3,1],[2]] => [4,2] => 100100 => 2

[[3,1],[1,1]] => [4,2] => 100100 => 2

[[2,2],[1],[1]] => [4,1,1] => 1000110 => 3

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

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

[[2,1,1],[1],[1]] => [4,1,1] => 1000110 => 3

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

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

[[1,1,1,1],[1],[1]] => [4,1,1] => 1000110 => 3

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

[[5],[1]] => [5,1] => 1000010 => 2

[[4,1],[1]] => [5,1] => 1000010 => 2

[[3,2],[1]] => [5,1] => 1000010 => 2

[[3,1,1],[1]] => [5,1] => 1000010 => 2

[[2,2,1],[1]] => [5,1] => 1000010 => 2

[[2,1,1,1],[1]] => [5,1] => 1000010 => 2

[[1,1,1,1,1],[1]] => [5,1] => 1000010 => 2

[[6]] => [6] => 1000000 => 1

[[5,1]] => [6] => 1000000 => 1

[[4,2]] => [6] => 1000000 => 1

[[4,1,1]] => [6] => 1000000 => 1

[[3,3]] => [6] => 1000000 => 1

[[3,2,1]] => [6] => 1000000 => 1

[[3,1,1,1]] => [6] => 1000000 => 1

[[2,2,2]] => [6] => 1000000 => 1

[[2,2,1,1]] => [6] => 1000000 => 1

[[2,1,1,1,1]] => [6] => 1000000 => 1

[[1,1,1,1,1,1]] => [6] => 1000000 => 1

[[1],[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1,1] => 11111110 => 7

[[2],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => 10111110 => 6

[[2],[2],[1],[1],[1]] => [2,2,1,1,1] => 1101110 => 5

[[2],[2],[2],[1]] => [2,2,2,1] => 111010 => 4

[[1,1],[1],[1],[1],[1],[1]] => [2,1,1,1,1,1] => 10111110 => 6

[[1,1],[1,1],[1],[1],[1]] => [2,2,1,1,1] => 1101110 => 5

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$F_{1} = q$

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

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

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

$F_{5} = 7\ q + 9\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5}$

$F_{6} = 11\ q + 18\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$

$F_{7} = 15\ q + 30\ q^{2} + 22\ q^{3} + 11\ q^{4} + 5\ q^{5} + 2\ q^{6} + q^{7}$

$F_{8} = 22\ q + 53\ q^{2} + 42\ q^{3} + 24\ q^{4} + 11\ q^{5} + 5\ q^{6} + 2\ q^{7} + q^{8}$

$F_{9} = 30\ q + 85\ q^{2} + 78\ q^{3} + 46\ q^{4} + 24\ q^{5} + 11\ q^{6} + 5\ q^{7} + 2\ q^{8} + q^{9}$

$F_{10} = 42\ q + 139\ q^{2} + 138\ q^{3} + 90\ q^{4} + 48\ q^{5} + 24\ q^{6} + 11\ q^{7} + 5\ q^{8} + 2\ q^{9} + q^{10}$

Description

The number of ones in a binary word.
This is also known as the Hamming weight of the word.

Map

to partition

Description

The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.

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.