Identifier
Values
([2],3) => [2] => 1
([1,1],3) => [1,1] => 1
([3,1],3) => [2,1] => 1
([2,1,1],3) => [1,1,1] => 1
([4,2],3) => [2,2] => 1
([3,1,1],3) => [2,1,1] => 1
([2,2,1,1],3) => [1,1,1,1] => 1
([5,3,1],3) => [2,2,1] => 1
([4,2,1,1],3) => [2,1,1,1] => 1
([3,2,2,1,1],3) => [1,1,1,1,1] => 1
([6,4,2],3) => [2,2,2] => 1
([5,3,1,1],3) => [2,2,1,1] => 1
([4,2,2,1,1],3) => [2,1,1,1,1] => 1
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => 1
([2],4) => [2] => 1
([1,1],4) => [1,1] => 1
([3],4) => [3] => 1
([2,1],4) => [2,1] => 1
([1,1,1],4) => [1,1,1] => 1
([4,1],4) => [3,1] => 1
([2,2],4) => [2,2] => 1
([3,1,1],4) => [2,1,1] => 1
([2,1,1,1],4) => [1,1,1,1] => 1
([5,2],4) => [3,2] => 2
([4,1,1],4) => [3,1,1] => 1
([3,2,1],4) => [2,2,1] => 1
([3,1,1,1],4) => [2,1,1,1] => 1
([2,2,1,1,1],4) => [1,1,1,1,1] => 1
([6,3],4) => [3,3] => 2
([5,2,1],4) => [3,2,1] => 2
([4,1,1,1],4) => [3,1,1,1] => 1
([4,2,2],4) => [2,2,2] => 1
([3,3,1,1],4) => [2,2,1,1] => 1
([3,2,1,1,1],4) => [2,1,1,1,1] => 1
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => 1
([2],5) => [2] => 1
([1,1],5) => [1,1] => 1
([3],5) => [3] => 1
([2,1],5) => [2,1] => 1
([1,1,1],5) => [1,1,1] => 1
([4],5) => [4] => 2
([3,1],5) => [3,1] => 1
([2,2],5) => [2,2] => 1
([2,1,1],5) => [2,1,1] => 1
([1,1,1,1],5) => [1,1,1,1] => 1
([5,1],5) => [4,1] => 2
([3,2],5) => [3,2] => 2
([4,1,1],5) => [3,1,1] => 1
([2,2,1],5) => [2,2,1] => 1
([3,1,1,1],5) => [2,1,1,1] => 1
([2,1,1,1,1],5) => [1,1,1,1,1] => 1
([6,2],5) => [4,2] => 5
([5,1,1],5) => [4,1,1] => 2
([3,3],5) => [3,3] => 2
([4,2,1],5) => [3,2,1] => 2
([4,1,1,1],5) => [3,1,1,1] => 1
([2,2,2],5) => [2,2,2] => 1
([3,2,1,1],5) => [2,2,1,1] => 1
([3,1,1,1,1],5) => [2,1,1,1,1] => 1
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => 1
([2],6) => [2] => 1
([1,1],6) => [1,1] => 1
([3],6) => [3] => 1
([2,1],6) => [2,1] => 1
([1,1,1],6) => [1,1,1] => 1
([4],6) => [4] => 2
([3,1],6) => [3,1] => 1
([2,2],6) => [2,2] => 1
([2,1,1],6) => [2,1,1] => 1
([1,1,1,1],6) => [1,1,1,1] => 1
([5],6) => [5] => 4
([4,1],6) => [4,1] => 2
([3,2],6) => [3,2] => 2
([3,1,1],6) => [3,1,1] => 1
([2,2,1],6) => [2,2,1] => 1
([2,1,1,1],6) => [2,1,1,1] => 1
([1,1,1,1,1],6) => [1,1,1,1,1] => 1
([6,1],6) => [5,1] => 4
([4,2],6) => [4,2] => 5
([5,1,1],6) => [4,1,1] => 2
([3,3],6) => [3,3] => 2
([3,2,1],6) => [3,2,1] => 2
([4,1,1,1],6) => [3,1,1,1] => 1
([2,2,2],6) => [2,2,2] => 1
([2,2,1,1],6) => [2,2,1,1] => 1
([3,1,1,1,1],6) => [2,1,1,1,1] => 1
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => 1
([7,2],6) => [5,2] => 12
([6,1,1],6) => [5,1,1] => 4
([4,3],6) => [4,3] => 10
([5,2,1],6) => [4,2,1] => 5
([5,1,1,1],6) => [4,1,1,1] => 2
([3,3,1],6) => [3,3,1] => 2
([3,2,2],6) => [3,2,2] => 3
([4,2,1,1],6) => [3,2,1,1] => 2
([4,1,1,1,1],6) => [3,1,1,1,1] => 1
([2,2,2,1],6) => [2,2,2,1] => 1
([3,2,1,1,1],6) => [2,2,1,1,1] => 1
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => 1
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => 1
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 number of ways to refine the partition into singletons.
For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$.
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.
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].