searching the database
Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000142
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1]
=> 0
[[1],[1]]
=> [1,1]
=> [2]
=> 1
[[2]]
=> [2]
=> [1,1]
=> 0
[[1,1]]
=> [2]
=> [1,1]
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> 0
[[2],[1]]
=> [2,1]
=> [2,1]
=> 1
[[1,1],[1]]
=> [2,1]
=> [2,1]
=> 1
[[3]]
=> [3]
=> [1,1,1]
=> 0
[[2,1]]
=> [3]
=> [1,1,1]
=> 0
[[1,1,1]]
=> [3]
=> [1,1,1]
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 0
[[2],[2]]
=> [2,2]
=> [2,2]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 0
[[1,1],[1,1]]
=> [2,2]
=> [2,2]
=> 2
[[3],[1]]
=> [3,1]
=> [2,1,1]
=> 1
[[2,1],[1]]
=> [3,1]
=> [2,1,1]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [2,1,1]
=> 1
[[4]]
=> [4]
=> [1,1,1,1]
=> 0
[[3,1]]
=> [4]
=> [1,1,1,1]
=> 0
[[2,2]]
=> [4]
=> [1,1,1,1]
=> 0
[[2,1,1]]
=> [4]
=> [1,1,1,1]
=> 0
[[1,1,1,1]]
=> [4]
=> [1,1,1,1]
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 0
[[3],[2]]
=> [3,2]
=> [2,2,1]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 0
[[2,1],[2]]
=> [3,2]
=> [2,2,1]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 0
[[1,1,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 2
[[4],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[3,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[2,2],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 1
[[5]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[4,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[3,2]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[3,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[2,2,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 0
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 2
Description
The number of even parts of a partition.
Matching statistic: St000992
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> 0
[[1],[1]]
=> [1,1]
=> [1]
=> 1
[[2]]
=> [2]
=> []
=> 0
[[1,1]]
=> [2]
=> []
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> 0
[[2],[1]]
=> [2,1]
=> [1]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> 1
[[3]]
=> [3]
=> []
=> 0
[[2,1]]
=> [3]
=> []
=> 0
[[1,1,1]]
=> [3]
=> []
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 0
[[2],[2]]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> 2
[[3],[1]]
=> [3,1]
=> [1]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> 1
[[4]]
=> [4]
=> []
=> 0
[[3,1]]
=> [4]
=> []
=> 0
[[2,2]]
=> [4]
=> []
=> 0
[[2,1,1]]
=> [4]
=> []
=> 0
[[1,1,1,1]]
=> [4]
=> []
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 0
[[3],[2]]
=> [3,2]
=> [2]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 0
[[2,1],[2]]
=> [3,2]
=> [2]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> 2
[[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
[[5]]
=> [5]
=> []
=> 0
[[4,1]]
=> [5]
=> []
=> 0
[[3,2]]
=> [5]
=> []
=> 0
[[3,1,1]]
=> [5]
=> []
=> 0
[[2,2,1]]
=> [5]
=> []
=> 0
[[2,1,1,1]]
=> [5]
=> []
=> 0
[[1,1,1,1,1]]
=> [5]
=> []
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000148
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> []
=> 0
[[1],[1]]
=> [1,1]
=> [1]
=> [1]
=> 1
[[2]]
=> [2]
=> []
=> []
=> 0
[[1,1]]
=> [2]
=> []
=> []
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[[2],[1]]
=> [2,1]
=> [1]
=> [1]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1]
=> [1]
=> 1
[[3]]
=> [3]
=> []
=> []
=> 0
[[2,1]]
=> [3]
=> []
=> []
=> 0
[[1,1,1]]
=> [3]
=> []
=> []
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[2],[2]]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,1],[1,1]]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[[3],[1]]
=> [3,1]
=> [1]
=> [1]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1]
=> [1]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1]
=> [1]
=> 1
[[4]]
=> [4]
=> []
=> []
=> 0
[[3,1]]
=> [4]
=> []
=> []
=> 0
[[2,2]]
=> [4]
=> []
=> []
=> 0
[[2,1,1]]
=> [4]
=> []
=> []
=> 0
[[1,1,1,1]]
=> [4]
=> []
=> []
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[3],[2]]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[2,1],[2]]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,1,1],[1,1]]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[[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
[[5]]
=> [5]
=> []
=> []
=> 0
[[4,1]]
=> [5]
=> []
=> []
=> 0
[[3,2]]
=> [5]
=> []
=> []
=> 0
[[3,1,1]]
=> [5]
=> []
=> []
=> 0
[[2,2,1]]
=> [5]
=> []
=> []
=> 0
[[2,1,1,1]]
=> [5]
=> []
=> []
=> 0
[[1,1,1,1,1]]
=> [5]
=> []
=> []
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
Description
The number of odd parts of a partition.
Matching statistic: St000149
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1]
=> [1]
=> 0
[[1],[1]]
=> [1,1]
=> [2]
=> [2]
=> 1
[[2]]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[[1,1]]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> [2,1]
=> 0
[[2],[1]]
=> [2,1]
=> [2,1]
=> [3]
=> 1
[[1,1],[1]]
=> [2,1]
=> [2,1]
=> [3]
=> 1
[[3]]
=> [3]
=> [1,1,1]
=> [1,1,1]
=> 0
[[2,1]]
=> [3]
=> [1,1,1]
=> [1,1,1]
=> 0
[[1,1,1]]
=> [3]
=> [1,1,1]
=> [1,1,1]
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 0
[[2],[2]]
=> [2,2]
=> [2,2]
=> [4]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 0
[[1,1],[1,1]]
=> [2,2]
=> [2,2]
=> [4]
=> 2
[[3],[1]]
=> [3,1]
=> [2,1,1]
=> [3,1]
=> 1
[[2,1],[1]]
=> [3,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [2,1,1]
=> [3,1]
=> 1
[[4]]
=> [4]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[[3,1]]
=> [4]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[[2,2]]
=> [4]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[[2,1,1]]
=> [4]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,1,1,1]]
=> [4]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> [2,2,1]
=> 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [3,2]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> [4,1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [3,2]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> [4,1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> 0
[[3],[2]]
=> [3,2]
=> [2,2,1]
=> [5]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> 0
[[2,1],[2]]
=> [3,2]
=> [2,2,1]
=> [5]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> [5]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> 0
[[1,1,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> [5]
=> 2
[[4],[1]]
=> [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[[3,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[[2,2],[1]]
=> [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[[5]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[4,1]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[3,2]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[3,1,1]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[2,2,1]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> [2,2,2]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> [2,2,1,1]
=> 0
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> [4,2]
=> 2
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000150
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1]
=> [1]
=> 0
[[1],[1]]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[[2]]
=> [2]
=> [1,1]
=> [2]
=> 0
[[1,1]]
=> [2]
=> [1,1]
=> [2]
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> [3]
=> 0
[[2],[1]]
=> [2,1]
=> [2,1]
=> [1,1,1]
=> 1
[[1,1],[1]]
=> [2,1]
=> [2,1]
=> [1,1,1]
=> 1
[[3]]
=> [3]
=> [1,1,1]
=> [2,1]
=> 0
[[2,1]]
=> [3]
=> [1,1,1]
=> [2,1]
=> 0
[[1,1,1]]
=> [3]
=> [1,1,1]
=> [2,1]
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> [3,1]
=> 0
[[2],[2]]
=> [2,2]
=> [2,2]
=> [1,1,1,1]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> [3,1]
=> 0
[[1,1],[1,1]]
=> [2,2]
=> [2,2]
=> [1,1,1,1]
=> 2
[[3],[1]]
=> [3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[[2,1],[1]]
=> [3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[[4]]
=> [4]
=> [1,1,1,1]
=> [4]
=> 0
[[3,1]]
=> [4]
=> [1,1,1,1]
=> [4]
=> 0
[[2,2]]
=> [4]
=> [1,1,1,1]
=> [4]
=> 0
[[2,1,1]]
=> [4]
=> [1,1,1,1]
=> [4]
=> 0
[[1,1,1,1]]
=> [4]
=> [1,1,1,1]
=> [4]
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> [5]
=> 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [2,2,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> [3,1,1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [2,2,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> [3,1,1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[[3],[2]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[[2,1],[2]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[[1,1,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 2
[[4],[1]]
=> [4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[[3,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[[2,2],[1]]
=> [4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[[5]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[4,1]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[3,2]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[3,1,1]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[2,2,1]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> [3,3]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> [5,1]
=> 0
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> 2
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St001232
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[[1],[1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1,1]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[1],[1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
[[2],[1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[2,1]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,1,1]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0
[[2],[2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0
[[1,1],[1,1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3],[1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[2,1],[1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1,1],[1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[3,1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[2,1,1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,1,1,1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0
[[3],[2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0
[[2,1],[2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,1],[1,1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0
[[1,1,1],[1,1]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[4],[1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[3,1],[1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[2,2],[1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[2,1,1],[1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[5]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[4,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[3,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[3,1,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[2,2,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[2,1,1,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,1,1,1,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 0
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 0
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[[3],[3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[[2,1],[2,1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[[1,1,1],[1,1,1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[4],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[[4],[2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[[3,1],[2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[3,1],[1,1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[[2,2],[2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[2,2],[1,1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[[2,1,1],[2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[2,1,1],[1,1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[[1,1,1,1],[1,1]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[5],[1]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[4,1],[1]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[3,2],[1]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[3,1,1],[1]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[3],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!