Identifier
Values
([2],3) => [2] => 100 => 010 => 2
([1,1],3) => [1,1] => 110 => 110 => 2
([3,1],3) => [3,1] => 10010 => 00110 => 4
([2,1,1],3) => [2,1,1] => 10110 => 01110 => 4
([4,2],3) => [4,2] => 100100 => 100010 => 6
([3,1,1],3) => [3,1,1] => 100110 => 001110 => 5
([2,2,1,1],3) => [2,2,1,1] => 110110 => 101110 => 6
([5,3,1],3) => [5,3,1] => 10010010 => 01000110 => 9
([4,2,1,1],3) => [4,2,1,1] => 10010110 => 10001110 => 8
([3,2,2,1,1],3) => [3,2,2,1,1] => 10110110 => 11001110 => 9
([6,4,2],3) => [6,4,2] => 100100100 => 001100010 => 12
([5,3,1,1],3) => [5,3,1,1] => 100100110 => 010001110 => 10
([4,2,2,1,1],3) => [4,2,2,1,1] => 100110110 => 110001110 => 10
([3,3,2,2,1,1],3) => [3,3,2,2,1,1] => 110110110 => 011101110 => 12
([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) => [4,1] => 100010 => 000110 => 5
([2,2],4) => [2,2] => 1100 => 1010 => 4
([3,1,1],4) => [3,1,1] => 100110 => 001110 => 5
([2,1,1,1],4) => [2,1,1,1] => 101110 => 011110 => 5
([5,2],4) => [5,2] => 1000100 => 1000010 => 7
([4,1,1],4) => [4,1,1] => 1000110 => 0001110 => 6
([3,2,1],4) => [3,2,1] => 101010 => 100110 => 6
([3,1,1,1],4) => [3,1,1,1] => 1001110 => 0011110 => 6
([2,2,1,1,1],4) => [2,2,1,1,1] => 1101110 => 1011110 => 7
([6,3],4) => [6,3] => 10001000 => 01000010 => 9
([5,2,1],4) => [5,2,1] => 10001010 => 10000110 => 8
([4,1,1,1],4) => [4,1,1,1] => 10001110 => 00011110 => 7
([4,2,2],4) => [4,2,2] => 1001100 => 1100010 => 8
([3,3,1,1],4) => [3,3,1,1] => 1100110 => 0101110 => 8
([3,2,1,1,1],4) => [3,2,1,1,1] => 10101110 => 10011110 => 8
([2,2,2,1,1,1],4) => [2,2,2,1,1,1] => 11101110 => 11011110 => 9
([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) => [5,1] => 1000010 => 0000110 => 6
([3,2],5) => [3,2] => 10100 => 10010 => 5
([4,1,1],5) => [4,1,1] => 1000110 => 0001110 => 6
([2,2,1],5) => [2,2,1] => 11010 => 10110 => 5
([3,1,1,1],5) => [3,1,1,1] => 1001110 => 0011110 => 6
([2,1,1,1,1],5) => [2,1,1,1,1] => 1011110 => 0111110 => 6
([6,2],5) => [6,2] => 10000100 => 10000010 => 8
([5,1,1],5) => [5,1,1] => 10000110 => 00001110 => 7
([3,3],5) => [3,3] => 11000 => 01010 => 6
([4,2,1],5) => [4,2,1] => 1001010 => 1000110 => 7
([4,1,1,1],5) => [4,1,1,1] => 10001110 => 00011110 => 7
([2,2,2],5) => [2,2,2] => 11100 => 11010 => 6
([3,2,1,1],5) => [3,2,1,1] => 1010110 => 1001110 => 7
([3,1,1,1,1],5) => [3,1,1,1,1] => 10011110 => 00111110 => 7
([2,2,1,1,1,1],5) => [2,2,1,1,1,1] => 11011110 => 10111110 => 8
([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) => [6,1] => 10000010 => 00000110 => 7
([4,2],6) => [4,2] => 100100 => 100010 => 6
([5,1,1],6) => [5,1,1] => 10000110 => 00001110 => 7
([3,3],6) => [3,3] => 11000 => 01010 => 6
([3,2,1],6) => [3,2,1] => 101010 => 100110 => 6
([4,1,1,1],6) => [4,1,1,1] => 10001110 => 00011110 => 7
([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) => [3,1,1,1,1] => 10011110 => 00111110 => 7
([2,1,1,1,1,1],6) => [2,1,1,1,1,1] => 10111110 => 01111110 => 7
([7,2],6) => [7,2] => 100000100 => 100000010 => 9
([6,1,1],6) => [6,1,1] => 100000110 => 000001110 => 8
([4,3],6) => [4,3] => 101000 => 010010 => 7
([5,2,1],6) => [5,2,1] => 10001010 => 10000110 => 8
([5,1,1,1],6) => [5,1,1,1] => 100001110 => 000011110 => 8
([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) => [4,2,1,1] => 10010110 => 10001110 => 8
([4,1,1,1,1],6) => [4,1,1,1,1] => 100011110 => 000111110 => 8
([2,2,2,1],6) => [2,2,2,1] => 111010 => 110110 => 7
([3,2,1,1,1],6) => [3,2,1,1,1] => 10101110 => 10011110 => 8
([3,1,1,1,1,1],6) => [3,1,1,1,1,1] => 100111110 => 001111110 => 8
([2,2,1,1,1,1,1],6) => [2,2,1,1,1,1,1] => 110111110 => 101111110 => 9
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
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
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
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
to partition
Description
Considers a core as a partition.
This embedding is graded and injective but not surjective on $k$-cores for a given parameter $k$, while it is surjective and neither graded nor injective on the collection of all cores.