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Matching statistic: St001491
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> 10 => 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> 110 => 1
[[2],[1]]
=> [2,1]
=> [1]
=> 10 => 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> 10 => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 110 => 1
[[2],[2]]
=> [2,2]
=> [2]
=> 100 => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 110 => 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> 100 => 1
[[3],[1]]
=> [3,1]
=> [1]
=> 10 => 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> 10 => 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> 10 => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 110 => 1
[[3],[2]]
=> [3,2]
=> [2]
=> 100 => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 110 => 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> 100 => 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> 100 => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 110 => 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> 100 => 1
[[4],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> 10 => 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[[3],[3]]
=> [3,3]
=> [3]
=> 1000 => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> 1000 => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> 1000 => 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 110 => 1
[[4],[2]]
=> [4,2]
=> [2]
=> 100 => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 110 => 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> 100 => 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> 100 => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 110 => 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> 100 => 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> 100 => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 110 => 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> 100 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let An=K[x]/(xn).
We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
Matching statistic: St000706
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000939
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001568
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000567
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}
for a partition \lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000620
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition \lambda \vdash n by the number of standard tableaux T of shape \lambda such that \min\big( \operatorname{Des}(T) \cup \{n\} \big) is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000929
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is 0 for partitions \lambda \neq 1^n and 1 for \lambda = 1^n.
Matching statistic: St001099
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees.
For a generating function f the associated formal group law is the symmetric function f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots), see [1].
This statistic records the coefficient of the monomial symmetric function m_\lambda times the product of the factorials of the parts of \lambda in the formal group law for leaf labelled binary trees, with generating function f(x) = 1-\sqrt{1-2x}, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that \lambda_i are colored i. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
Matching statistic: St001100
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001100: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001100: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[[1],[1]]
=> [1,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> []
=> ? = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2],[2]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> []
=> ? = 1 - 1
[[3],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1],[2]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> []
=> ? = 1 - 1
[[4],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1]
=> []
=> ? = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[3],[3]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[2,1],[2,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? = 0 - 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [3]
=> []
=> ? = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[4],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[3,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[3,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,2],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,2],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[2,1,1],[2]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[2,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[[1,1,1,1],[1,1]]
=> [4,2]
=> [2]
=> []
=> ? = 1 - 1
[[5],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[4,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,2],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3,1,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[2,2,1],[1]]
=> [5,1]
=> [1]
=> []
=> ? = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[[6],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[5,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,2],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[4,1,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,3],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[3,2,1],[1],[1],[1]]
=> [6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees.
For a generating function f the associated formal group law is the symmetric function f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots), see [1].
This statistic records the coefficient of the monomial symmetric function m_\lambda times the product of the factorials of the parts of \lambda in the formal group law for leaf labelled binary trees, whose generating function is the reversal of f^{(-1)}(x) = 1+2x-\exp(x), see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that \lambda_i are colored i. This statistic gives the number of rooted trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
The following 1 statistic also match your data. Click on any of them to see the details.
St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees.
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