Identifier
Values
([2],3) => [2] => 100 => 010 => 2
([1,1],3) => [1,1] => 110 => 110 => 2
([3,1],3) => [2,1] => 1010 => 0110 => 3
([2,1,1],3) => [1,1,1] => 1110 => 1110 => 3
([4,2],3) => [2,2] => 1100 => 1010 => 4
([3,1,1],3) => [2,1,1] => 10110 => 01110 => 4
([2,2,1,1],3) => [1,1,1,1] => 11110 => 11110 => 4
([5,3,1],3) => [2,2,1] => 11010 => 10110 => 5
([4,2,1,1],3) => [2,1,1,1] => 101110 => 011110 => 5
([3,2,2,1,1],3) => [1,1,1,1,1] => 111110 => 111110 => 5
([6,4,2],3) => [2,2,2] => 11100 => 11010 => 6
([5,3,1,1],3) => [2,2,1,1] => 110110 => 101110 => 6
([4,2,2,1,1],3) => [2,1,1,1,1] => 1011110 => 0111110 => 6
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => 1111110 => 1111110 => 6
([2],4) => [2] => 100 => 010 => 2
([1,1],4) => [1,1] => 110 => 110 => 2
([3],4) => [3] => 1000 => 0010 => 3
([2,1],4) => [2,1] => 1010 => 0110 => 3
([1,1,1],4) => [1,1,1] => 1110 => 1110 => 3
([4,1],4) => [3,1] => 10010 => 00110 => 4
([2,2],4) => [2,2] => 1100 => 1010 => 4
([3,1,1],4) => [2,1,1] => 10110 => 01110 => 4
([2,1,1,1],4) => [1,1,1,1] => 11110 => 11110 => 4
([5,2],4) => [3,2] => 10100 => 10010 => 5
([4,1,1],4) => [3,1,1] => 100110 => 001110 => 5
([3,2,1],4) => [2,2,1] => 11010 => 10110 => 5
([3,1,1,1],4) => [2,1,1,1] => 101110 => 011110 => 5
([2,2,1,1,1],4) => [1,1,1,1,1] => 111110 => 111110 => 5
([6,3],4) => [3,3] => 11000 => 01010 => 6
([5,2,1],4) => [3,2,1] => 101010 => 100110 => 6
([4,1,1,1],4) => [3,1,1,1] => 1001110 => 0011110 => 6
([4,2,2],4) => [2,2,2] => 11100 => 11010 => 6
([3,3,1,1],4) => [2,2,1,1] => 110110 => 101110 => 6
([3,2,1,1,1],4) => [2,1,1,1,1] => 1011110 => 0111110 => 6
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => 1111110 => 1111110 => 6
([2],5) => [2] => 100 => 010 => 2
([1,1],5) => [1,1] => 110 => 110 => 2
([3],5) => [3] => 1000 => 0010 => 3
([2,1],5) => [2,1] => 1010 => 0110 => 3
([1,1,1],5) => [1,1,1] => 1110 => 1110 => 3
([4],5) => [4] => 10000 => 00010 => 4
([3,1],5) => [3,1] => 10010 => 00110 => 4
([2,2],5) => [2,2] => 1100 => 1010 => 4
([2,1,1],5) => [2,1,1] => 10110 => 01110 => 4
([1,1,1,1],5) => [1,1,1,1] => 11110 => 11110 => 4
([5,1],5) => [4,1] => 100010 => 000110 => 5
([3,2],5) => [3,2] => 10100 => 10010 => 5
([4,1,1],5) => [3,1,1] => 100110 => 001110 => 5
([2,2,1],5) => [2,2,1] => 11010 => 10110 => 5
([3,1,1,1],5) => [2,1,1,1] => 101110 => 011110 => 5
([2,1,1,1,1],5) => [1,1,1,1,1] => 111110 => 111110 => 5
([6,2],5) => [4,2] => 100100 => 100010 => 6
([5,1,1],5) => [4,1,1] => 1000110 => 0001110 => 6
([3,3],5) => [3,3] => 11000 => 01010 => 6
([4,2,1],5) => [3,2,1] => 101010 => 100110 => 6
([4,1,1,1],5) => [3,1,1,1] => 1001110 => 0011110 => 6
([2,2,2],5) => [2,2,2] => 11100 => 11010 => 6
([3,2,1,1],5) => [2,2,1,1] => 110110 => 101110 => 6
([3,1,1,1,1],5) => [2,1,1,1,1] => 1011110 => 0111110 => 6
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => 1111110 => 1111110 => 6
([2],6) => [2] => 100 => 010 => 2
([1,1],6) => [1,1] => 110 => 110 => 2
([3],6) => [3] => 1000 => 0010 => 3
([2,1],6) => [2,1] => 1010 => 0110 => 3
([1,1,1],6) => [1,1,1] => 1110 => 1110 => 3
([4],6) => [4] => 10000 => 00010 => 4
([3,1],6) => [3,1] => 10010 => 00110 => 4
([2,2],6) => [2,2] => 1100 => 1010 => 4
([2,1,1],6) => [2,1,1] => 10110 => 01110 => 4
([1,1,1,1],6) => [1,1,1,1] => 11110 => 11110 => 4
([5],6) => [5] => 100000 => 000010 => 5
([4,1],6) => [4,1] => 100010 => 000110 => 5
([3,2],6) => [3,2] => 10100 => 10010 => 5
([3,1,1],6) => [3,1,1] => 100110 => 001110 => 5
([2,2,1],6) => [2,2,1] => 11010 => 10110 => 5
([2,1,1,1],6) => [2,1,1,1] => 101110 => 011110 => 5
([1,1,1,1,1],6) => [1,1,1,1,1] => 111110 => 111110 => 5
([6,1],6) => [5,1] => 1000010 => 0000110 => 6
([4,2],6) => [4,2] => 100100 => 100010 => 6
([5,1,1],6) => [4,1,1] => 1000110 => 0001110 => 6
([3,3],6) => [3,3] => 11000 => 01010 => 6
([3,2,1],6) => [3,2,1] => 101010 => 100110 => 6
([4,1,1,1],6) => [3,1,1,1] => 1001110 => 0011110 => 6
([2,2,2],6) => [2,2,2] => 11100 => 11010 => 6
([2,2,1,1],6) => [2,2,1,1] => 110110 => 101110 => 6
([3,1,1,1,1],6) => [2,1,1,1,1] => 1011110 => 0111110 => 6
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => 1111110 => 1111110 => 6
([7,2],6) => [5,2] => 1000100 => 1000010 => 7
([6,1,1],6) => [5,1,1] => 10000110 => 00001110 => 7
([4,3],6) => [4,3] => 101000 => 010010 => 7
([5,2,1],6) => [4,2,1] => 1001010 => 1000110 => 7
([5,1,1,1],6) => [4,1,1,1] => 10001110 => 00011110 => 7
([3,3,1],6) => [3,3,1] => 110010 => 010110 => 7
([3,2,2],6) => [3,2,2] => 101100 => 110010 => 7
([4,2,1,1],6) => [3,2,1,1] => 1010110 => 1001110 => 7
([4,1,1,1,1],6) => [3,1,1,1,1] => 10011110 => 00111110 => 7
([2,2,2,1],6) => [2,2,2,1] => 111010 => 110110 => 7
([3,2,1,1,1],6) => [2,2,1,1,1] => 1101110 => 1011110 => 7
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => 10111110 => 01111110 => 7
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => 11111110 => 11111110 => 7
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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 bounded partition
Description
The (k-1)-bounded partition of a k-core.
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
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.