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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St000659
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001035
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St001036
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[1,1],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,2],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,4],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[[1,1,1],[2,2,2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001621
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001630
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,2],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[3,3],[4]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 0 + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 0 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 0 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 2
[[1],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[4],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[5],[6]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,2],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 0 + 2
[[1,3],[5]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1],[3]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[3]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1],[4]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[4]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[3],[4]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,2],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,3],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[1,1,1],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,2],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,2],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1],[2,2]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[3],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[4],[5]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,4],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,4],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,4],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,2],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,3],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,4],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[2,4],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[3,3],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[3,4],[4]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1],[2],[4]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> 1110 => 1110 => 2 = 1 + 1
[[1,1,1],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,2],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,3],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,2],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,3],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,3],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,2],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,3],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1],[2,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,1],[3,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,2],[3,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[2,2],[3,3]]
=> [2,2]
=> 1100 => 0110 => 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> 10110 => 11010 => ? = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> 10110 => 11010 => ? = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> 10110 => 11010 => ? = 1 + 1
[[1,1,1,1],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,1,1,2],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,1,2,2],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,2,2,2],[2]]
=> [4,1]
=> 100010 => 100100 => ? = 0 + 1
[[1,1,1],[2,2]]
=> [3,2]
=> 10100 => 01100 => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,2]
=> 10100 => 01100 => ? = 1 + 1
[[1],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[2],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[3],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[4],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[5],[6]]
=> [1,1]
=> 110 => 110 => 1 = 0 + 1
[[1,1],[5]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,2],[5]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,5],[2]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,3],[5]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,5],[3]]
=> [2,1]
=> 1010 => 1100 => 1 = 0 + 1
[[1,1,1],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,2],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,1,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,2],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,2,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,4,4],[2]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,3,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,4,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[1,4,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,2],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,2,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,3],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,3,4],[4]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
[[2,4,4],[3]]
=> [3,1]
=> 10010 => 10100 => ? = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-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 $M_S$. 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: St001207
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[3]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[3],[4]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[1,1,1],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[3],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[4],[5]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,4],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,2],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[2,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[3,3],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[3,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[[1,1,1],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,2],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[[1,1,1,1],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,1,1,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,1,2,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,2,2,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 0 + 2
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 2
[[1],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[2],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[3],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[4],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[5],[6]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
[[1,1],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,2],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,5],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,3],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,5],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
[[1,1,1],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,2],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,1,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,2],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,2,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,4,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,4,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[1,4,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,2],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,2,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
[[2,4,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 0 + 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001857The number of edges in the reduced word graph of a signed permutation. St000166The depth minus 1 of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St001645The pebbling number of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St000454The largest eigenvalue of a graph if it is integral. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000422The energy of a graph, if it is integral. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001407The number of minimal entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St000103The sum of the entries of a semistandard tableau.
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