- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤ

Values

[[2,1],[1]] => [1] => 10 => 10 => 1

[[3,1],[1]] => [1] => 10 => 10 => 1

[[2,2],[1]] => [1] => 10 => 10 => 1

[[3,2],[2]] => [2] => 100 => 010 => 2

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

[[2,1,1],[1]] => [1] => 10 => 10 => 1

[[3,2,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,1],[1]] => [1] => 10 => 10 => 1

[[3,2],[1]] => [1] => 10 => 10 => 1

[[4,2],[2]] => [2] => 100 => 010 => 2

[[3,2,1],[1,1]] => [1,1] => 110 => 110 => 2

[[3,1,1],[1]] => [1] => 10 => 10 => 1

[[4,2,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[3,3],[2]] => [2] => 100 => 010 => 2

[[4,3],[3]] => [3] => 1000 => 0010 => 3

[[2,2,1],[1]] => [1] => 10 => 10 => 1

[[3,3,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[3,2,1],[2]] => [2] => 100 => 010 => 2

[[4,3,1],[3,1]] => [3,1] => 10010 => 00110 => 4

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

[[3,3,2],[2,2]] => [2,2] => 1100 => 1010 => 4

[[3,2,2],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,3,2],[3,2]] => [3,2] => 10100 => 10010 => 5

[[2,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 1110 => 3

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

[[3,3,2,1],[2,2,1]] => [2,2,1] => 11010 => 10110 => 5

[[2,1,1,1],[1]] => [1] => 10 => 10 => 1

[[3,2,2,1],[2,1,1]] => [2,1,1] => 10110 => 01110 => 4

[[3,2,1,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,3,2,1],[3,2,1]] => [3,2,1] => 101010 => 100110 => 6

[[5,1],[1]] => [1] => 10 => 10 => 1

[[4,2],[1]] => [1] => 10 => 10 => 1

[[5,2],[2]] => [2] => 100 => 010 => 2

[[4,2,1],[1,1]] => [1,1] => 110 => 110 => 2

[[4,1,1],[1]] => [1] => 10 => 10 => 1

[[5,2,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[3,3],[1]] => [1] => 10 => 10 => 1

[[4,3],[2]] => [2] => 100 => 010 => 2

[[5,3],[3]] => [3] => 1000 => 0010 => 3

[[3,3,1],[1,1]] => [1,1] => 110 => 110 => 2

[[3,2,1],[1]] => [1] => 10 => 10 => 1

[[4,3,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,2,1],[2]] => [2] => 100 => 010 => 2

[[5,3,1],[3,1]] => [3,1] => 10010 => 00110 => 4

[[3,2,2],[1,1]] => [1,1] => 110 => 110 => 2

[[4,3,2],[2,2]] => [2,2] => 1100 => 1010 => 4

[[4,2,2],[2,1]] => [2,1] => 1010 => 0110 => 3

[[5,3,2],[3,2]] => [3,2] => 10100 => 10010 => 5

[[3,2,2,1],[1,1,1]] => [1,1,1] => 1110 => 1110 => 3

[[3,2,1,1],[1,1]] => [1,1] => 110 => 110 => 2

[[4,3,2,1],[2,2,1]] => [2,2,1] => 11010 => 10110 => 5

[[3,1,1,1],[1]] => [1] => 10 => 10 => 1

[[4,2,2,1],[2,1,1]] => [2,1,1] => 10110 => 01110 => 4

[[4,2,1,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[5,3,2,1],[3,2,1]] => [3,2,1] => 101010 => 100110 => 6

[[4,4],[3]] => [3] => 1000 => 0010 => 3

[[5,4],[4]] => [4] => 10000 => 00010 => 4

[[3,3,1],[2]] => [2] => 100 => 010 => 2

[[4,4,1],[3,1]] => [3,1] => 10010 => 00110 => 4

[[4,3,1],[3]] => [3] => 1000 => 0010 => 3

[[5,4,1],[4,1]] => [4,1] => 100010 => 000110 => 5

[[2,2,2],[1]] => [1] => 10 => 10 => 1

[[3,3,2],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,4,2],[3,2]] => [3,2] => 10100 => 10010 => 5

[[3,2,2],[2]] => [2] => 100 => 010 => 2

[[4,3,2],[3,1]] => [3,1] => 10010 => 00110 => 4

[[5,4,2],[4,2]] => [4,2] => 100100 => 100010 => 6

[[2,2,1,1],[1]] => [1] => 10 => 10 => 1

[[3,3,2,1],[2,1,1]] => [2,1,1] => 10110 => 01110 => 4

[[3,3,1,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,4,2,1],[3,2,1]] => [3,2,1] => 101010 => 100110 => 6

[[3,2,1,1],[2]] => [2] => 100 => 010 => 2

[[4,3,2,1],[3,1,1]] => [3,1,1] => 100110 => 001110 => 5

[[4,3,1,1],[3,1]] => [3,1] => 10010 => 00110 => 4

[[5,4,2,1],[4,2,1]] => [4,2,1] => 1001010 => 1000110 => 7

[[3,3,3],[2,2]] => [2,2] => 1100 => 1010 => 4

[[4,4,3],[3,3]] => [3,3] => 11000 => 01010 => 6

[[4,3,3],[3,2]] => [3,2] => 10100 => 10010 => 5

[[5,4,3],[4,3]] => [4,3] => 101000 => 010010 => 7

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

[[3,3,3,1],[2,2,1]] => [2,2,1] => 11010 => 10110 => 5

[[3,3,2,1],[2,2]] => [2,2] => 1100 => 1010 => 4

[[4,4,3,1],[3,3,1]] => [3,3,1] => 110010 => 010110 => 7

[[3,2,2,1],[2,1]] => [2,1] => 1010 => 0110 => 3

[[4,3,3,1],[3,2,1]] => [3,2,1] => 101010 => 100110 => 6

[[4,3,2,1],[3,2]] => [3,2] => 10100 => 10010 => 5

[[5,4,3,1],[4,3,1]] => [4,3,1] => 1010010 => 0100110 => 8

[[2,2,2,2],[1,1,1]] => [1,1,1] => 1110 => 1110 => 3

[[3,3,3,2],[2,2,2]] => [2,2,2] => 11100 => 11010 => 6

[[3,3,2,2],[2,2,1]] => [2,2,1] => 11010 => 10110 => 5

[[4,4,3,2],[3,3,2]] => [3,3,2] => 110100 => 011010 => 8

[[3,2,2,2],[2,1,1]] => [2,1,1] => 10110 => 01110 => 4

[[4,3,3,2],[3,2,2]] => [3,2,2] => 101100 => 110010 => 7

[[4,3,2,2],[3,2,1]] => [3,2,1] => 101010 => 100110 => 6

[[5,4,3,2],[4,3,2]] => [4,3,2] => 1010100 => 0110010 => 9

[[2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 11110 => 11110 => 4

[[2,2,2,1,1],[1,1,1]] => [1,1,1] => 1110 => 1110 => 3

[[3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => 111010 => 110110 => 7

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

[[3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => 110110 => 101110 => 6

[[3,3,2,1,1],[2,2,1]] => [2,2,1] => 11010 => 10110 => 5

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searching the database for the individual values of this statistic

Description

The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.

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.

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00096Foata bijection.

Map

inner shape

Description

The inner shape of a skew partition.