searching the database
Your data matches 256 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: St000065
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000065: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> 0
[[0,1],[1,0]]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St000074
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [[2,1],[2]]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [[2,1],[1]]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [[3,2,1],[3,2],[3]]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [[3,2,1],[3,2],[2]]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [[3,2,1],[3,1],[3]]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [[3,2,1],[3,1],[2]]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [[3,2,1],[2,1],[2]]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [[3,2,1],[3,1],[1]]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [[3,2,1],[2,1],[1]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[4]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[3]]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[4]]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[3]]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[2]]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[2]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[4]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[3]]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[4]]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[3]]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[2]]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[2]]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[4]]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[3]]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[4]]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[3]]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[2,1],[2]]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[2]]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[1]]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[1]]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[1]]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[1]]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[2,1],[1]]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[1]]
=> 0
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000149
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000256
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000257
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000478
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$
\sum_{\pi\in GG_1(r)} w_1(\pi)
$$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St001163
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00006: Alternating sign matrices —gyration⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001163: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001163: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[0,1],[1,0]]
=> [1,1,0,0]
=> 0
[[0,1],[1,0]]
=> [[1,0],[0,1]]
=> [1,0,1,0]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 0
Description
The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra.
Matching statistic: St001392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001714
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> 0
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> 0
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
The following 246 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000667The greatest common divisor of the parts of the partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001571The Cartan determinant of the integer partition. St000145The Dyson rank of a partition. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000731The number of double exceedences of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001413Half the length of the longest even length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000160The multiplicity of the smallest part of a partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000920The logarithmic height of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001732The number of peaks visible from the left. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000439The position of the first down step of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000219The number of occurrences of the pattern 231 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000264The girth of a graph, which is not a tree. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000022The number of fixed points of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St000153The number of adjacent cycles of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000871The number of very big ascents of a permutation. St000842The breadth of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000709The number of occurrences of 14-2-3 or 14-3-2. St000077The number of boxed and circled entries. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000779The tier of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000873The aix statistic of a permutation. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001586The number of odd parts smaller than the largest even part in an integer partition. St001715The number of non-records in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000260The radius of a connected graph. St000487The length of the shortest cycle of a permutation. St000570The Edelman-Greene number of a permutation. St000618The number of self-evacuating tableaux of given shape. St000694The number of affine bounded permutations that project to a given permutation. St000711The number of big exceedences of a permutation. St001162The minimum jump of a permutation. St001432The order dimension of the partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001741The largest integer such that all patterns of this size are contained in the permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St000068The number of minimal elements in a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001490The number of connected components of a skew partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St000894The trace of an alternating sign matrix. St001260The permanent of an alternating sign matrix. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001902The number of potential covers of a poset. St001964The interval resolution global dimension of a poset. St000100The number of linear extensions of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000527The width of the poset. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001613The binary logarithm of the size of the center of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001779The order of promotion on the set of linear extensions of a poset. St001881The number of factors of a lattice as a Cartesian product of lattices. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.
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!