searching the database
Your data matches 81 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: St000510
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
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: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 40%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 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],[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
[[2,2],[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
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 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]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 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]
=> ? = 2 + 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,2],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,3],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,3],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2,2],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[2,2,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[2,2,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 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]
=> ? = 2 + 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]
=> ? = 2 + 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]
=> ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,2,2],[2,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 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]
=> ? = 0 + 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]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[3,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[4,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[5,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[2,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[4,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,4],[2,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[5,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[3,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[4,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 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: St001771
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 40%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => 2 = 1 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => 2 = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => [4,5,2,1,-3] => ? = 1 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => [4,2,5,1,-3] => ? = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [4,2,1,5,-3] => ? = 1 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => [2,4,5,1,-3] => ? = 1 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => ? = 1 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => ? = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [5,2,3,1,-4] => ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => ? = 2 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 1 + 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 1 + 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 1 + 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 1 + 1
[[1,2,4],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,2,3],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,2,4],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,2,4],[4,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
[[1,3,3],[4,4]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 1 + 1
[[1,3,4],[4,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 1 + 1
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001207
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,4,2] => 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [1,2,4,5,3] => ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [1,3,4,2,5] => ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ? = 1 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ? = 1 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,1,2,5,4] => ? = 1 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,5,2,4,3] => ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [1,5,3,2,4] => ? = 2 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [1,4,5,2,3] => ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,3,6,4,5] => ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [1,2,5,3,4,6] => ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [1,4,2,3,5,6] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [1,2,6,3,4,5] => ? = 0 + 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [1,2,4,5,3] => ? = 1 + 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [1,2,4,5,3] => ? = 1 + 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [1,2,4,5,3] => ? = 1 + 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [1,4,2,3,5] => ? = 1 + 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [1,3,4,2,5] => ? = 1 + 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [1,2,5,3,4] => ? = 1 + 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [1,3,4,2,5] => ? = 1 + 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [1,2,4,5,3] => ? = 1 + 1
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 $K[x]/(x^n)$.
Matching statistic: St001937
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001937: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001937: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1,1],[2,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 2
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,4,2,3,1,6] => [5,4,2,3,1,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 0
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
Description
The size of the center of a parking function.
The center of a parking function $p_1,\dots,p_n$ is the longest subsequence $a_1,\dots,a_k$ such that $a_i\leq i$.
Matching statistic: St001896
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [4,6,1,2,3,5] => ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,5,1,2,4,6] => ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => ? = 0 + 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 + 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 + 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1 + 1
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St001946
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 1 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 2 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,6,3] => ? = 0 + 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 + 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 + 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 + 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 + 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 + 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 + 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1 + 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 + 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 + 1
Description
The number of descents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St001719
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[[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
[[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
[[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
[[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],[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
[[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
[[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
[[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
[[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],[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
[[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
[[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
[[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
[[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
[[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
[[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
[[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
[[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],[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],[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
[[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
[[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
[[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
[[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
[[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
[[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
[[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
[[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
[[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
[[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],[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],[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
[[2,3],[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
[[2,3],[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
[[3,3],[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
[[1,1],[2],[4]]
=> [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
[[1,1],[3],[4]]
=> [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
[[1,2],[2],[4]]
=> [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
[[1,2],[3],[4]]
=> [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
[[1,3],[2],[4]]
=> [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
[[1,4],[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
[[1,4],[2],[4]]
=> [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
[[1,3],[3],[4]]
=> [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
[[1,4],[3],[4]]
=> [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
[[2,2],[3],[4]]
=> [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
[[2,3],[3],[4]]
=> [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
[[2,4],[3],[4]]
=> [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
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 2
[[1,1,1],[2,3]]
=> [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
[[1,1,1],[3,3]]
=> [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
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1
[[1,1,3],[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
[[1,1,2],[3,3]]
=> [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
[[1,1,3],[2,3]]
=> [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
[[1,1,3],[3,3]]
=> [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
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 1
[[1,2,2],[3,3]]
=> [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
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 1
[[1,2,3],[3,3]]
=> [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
[[2,2,2],[3,3]]
=> [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
[[2,2,3],[3,3]]
=> [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
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? = 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 2
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ? = 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 0
[[1],[2],[6]]
=> [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
[[1],[3],[6]]
=> [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
[[1],[4],[6]]
=> [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
[[1],[5],[6]]
=> [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],[6]]
=> [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],[4],[6]]
=> [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],[5],[6]]
=> [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
[[3],[4],[6]]
=> [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
[[3],[5],[6]]
=> [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
[[4],[5],[6]]
=> [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
[[1,1],[2,5]]
=> [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
[[1,1],[3,5]]
=> [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
[[1,1],[4,5]]
=> [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
[[1,1],[5,5]]
=> [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
[[1,1],[2],[5]]
=> [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
[[1,1],[3],[5]]
=> [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
[[1,1],[4],[5]]
=> [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
[[1,2],[2],[5]]
=> [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
[[1,2],[3],[5]]
=> [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
[[1,3],[2],[5]]
=> [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
[[1,2],[4],[5]]
=> [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
[[1,4],[2],[5]]
=> [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
[[1,3],[3],[5]]
=> [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
[[1,3],[4],[5]]
=> [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
[[1,4],[3],[5]]
=> [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
[[1,4],[4],[5]]
=> [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
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001845
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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 - 1
[[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 - 1
[[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)
=> 0 = 1 - 1
[[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 - 1
[[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 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[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)
=> 0 = 1 - 1
[[2,3],[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)
=> 0 = 1 - 1
[[2,3],[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)
=> 0 = 1 - 1
[[3,3],[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)
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [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 - 1
[[1,1],[3],[4]]
=> [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 - 1
[[1,2],[2],[4]]
=> [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 - 1
[[1,2],[3],[4]]
=> [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 - 1
[[1,3],[2],[4]]
=> [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 - 1
[[1,4],[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)
=> 0 = 1 - 1
[[1,4],[2],[4]]
=> [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)
=> 0 = 1 - 1
[[1,3],[3],[4]]
=> [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 - 1
[[1,4],[3],[4]]
=> [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)
=> 0 = 1 - 1
[[2,2],[3],[4]]
=> [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 - 1
[[2,3],[3],[4]]
=> [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 - 1
[[2,4],[3],[4]]
=> [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)
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 2 - 1
[[1,1,1],[2,3]]
=> [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 - 1
[[1,1,1],[3,3]]
=> [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 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1 - 1
[[1,1,3],[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 - 1
[[1,1,2],[3,3]]
=> [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 - 1
[[1,1,3],[2,3]]
=> [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 - 1
[[1,1,3],[3,3]]
=> [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 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 1 - 1
[[1,2,2],[3,3]]
=> [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 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 0 = 1 - 1
[[1,2,3],[3,3]]
=> [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 - 1
[[2,2,2],[3,3]]
=> [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 - 1
[[2,2,3],[3,3]]
=> [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 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 1 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 1 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 1 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 1 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 1 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 1 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? = 2 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 2 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ? = 2 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 0 - 1
[[1],[2],[6]]
=> [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)
=> 0 = 1 - 1
[[1],[3],[6]]
=> [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)
=> 0 = 1 - 1
[[1],[4],[6]]
=> [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)
=> 0 = 1 - 1
[[1],[5],[6]]
=> [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)
=> 0 = 1 - 1
[[2],[3],[6]]
=> [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)
=> 0 = 1 - 1
[[2],[4],[6]]
=> [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)
=> 0 = 1 - 1
[[2],[5],[6]]
=> [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)
=> 0 = 1 - 1
[[3],[4],[6]]
=> [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)
=> 0 = 1 - 1
[[3],[5],[6]]
=> [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)
=> 0 = 1 - 1
[[4],[5],[6]]
=> [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)
=> 0 = 1 - 1
[[1,1],[2,5]]
=> [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)
=> 0 = 1 - 1
[[1,1],[3,5]]
=> [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)
=> 0 = 1 - 1
[[1,1],[4,5]]
=> [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)
=> 0 = 1 - 1
[[1,1],[5,5]]
=> [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)
=> 0 = 1 - 1
[[1,1],[2],[5]]
=> [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 - 1
[[1,1],[3],[5]]
=> [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 - 1
[[1,1],[4],[5]]
=> [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 - 1
[[1,2],[2],[5]]
=> [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 - 1
[[1,2],[3],[5]]
=> [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 - 1
[[1,3],[2],[5]]
=> [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 - 1
[[1,2],[4],[5]]
=> [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 - 1
[[1,4],[2],[5]]
=> [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 - 1
[[1,3],[3],[5]]
=> [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 - 1
[[1,3],[4],[5]]
=> [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 - 1
[[1,4],[3],[5]]
=> [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 - 1
[[1,4],[4],[5]]
=> [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 - 1
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001636
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 40%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 40%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1],[2],[4]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[1],[2],[5]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[2,3],[3,4]]
=> [2,4,1,3] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[2,3],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[3,3],[4,4]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 2 + 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 2
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 2
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 2
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 2 + 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 2
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [3,4,5,2,1] => ([(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)
=> ? = 2 + 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,3,4,6,5,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 1 + 2
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [2,3,5,4,1,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ? = 1 + 2
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,3,1,5,6] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 1 + 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 0 + 2
[[1],[2],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1],[3],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1],[4],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1],[5],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[2],[3],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[2],[4],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[2],[5],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[3],[4],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[3],[5],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[4],[5],[6]]
=> [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1,1],[2,5]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[3,5]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[4,5]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[5,5]]
=> [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[[1,1],[2],[5]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,1],[3],[5]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,1],[4],[5]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,2],[2],[5]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,2],[3],[5]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,3],[2],[5]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,2],[4],[5]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,4],[2],[5]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,3],[3],[5]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,3],[4],[5]]
=> [4,3,1,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,4],[3],[5]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
[[1,4],[4],[5]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 2
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000782The indicator function of whether a given perfect matching is an L & P matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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 $eA$. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001490The number of connected components of a skew partition. St001624The breadth of a lattice. 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. St000910The number of maximal chains of minimal length in a poset. St000307The number of rowmotion orbits of a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001754The number of tolerances of a finite lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000080The rank of the poset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001625The Möbius invariant of a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001964The interval resolution global dimension of a poset. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001645The pebbling number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000189The number of elements in the poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000400The path length of an ordered tree. St000180The number of chains of a poset. St000529The number of permutations whose descent word is the given binary word. St000100The number of linear extensions of a poset. St000416The number of inequivalent increasing trees of an ordered tree. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001909The number of interval-closed sets of a poset. St000410The tree factorial of an ordered tree. St000634The number of endomorphisms of a poset. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.
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!