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Your data matches 101 different statistics following compositions of up to 3 maps.
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Matching statistic: St001568
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Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: 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
St001568: 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]
=> 2
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001732
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000523
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000523: Ordered trees ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000523: Ordered trees ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 0 = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 1 - 1
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[],[],[]],[],[],[],[]]
=> ? = 1 - 1
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[],[],[]],[],[],[],[]]
=> ? = 1 - 1
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[],[],[]],[],[],[],[]]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[[],[],[],[]],[],[],[]]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1,2,2,2],[3,3],[4]]
=> [10,8,9,1,2,3,4,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2,2,2,2],[3,3],[4]]
=> [10,8,9,1,2,3,4,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1,2,2,3],[3,3],[4]]
=> [10,7,8,1,2,3,4,5,6,9] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ?
=> ? = 1 - 1
Description
The number of 2-protected nodes of a rooted tree.
This is the number of nodes with minimal distance two to a leaf. The number of trees with no 2-protected nodes is [[oeis:A143363]].
Matching statistic: St001022
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1,1,1,1],[2,2]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1,2],[2,2]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 1 - 1
[[1,1,1,2,2,2],[2,2]]
=> [4,5,1,2,3,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[2,2,2]]
=> [6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[2,2,2]]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[2,2,2]]
=> [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2,2]]
=> [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ?
=> ? = 1 - 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001113
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,1,1,2],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,1,1,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,1,2,2],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,1,2,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,2,2,2],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,2,2,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,2,3,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,3,3,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,2,2,2,2],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,2,2,2,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,2,2,3,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,2,3,3,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,3,3,3,3],[2],[3]]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[1,1,1,1,1,1],[2,2]]
=> [6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1,1,2],[2,2]]
=> [6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2,2,2],[2,2]]
=> [6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,2,2,2],[2,2]]
=> [6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Matching statistic: St000665
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0 = 1 - 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,1],[2,2,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,1],[2,3,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,1],[3,3,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,2],[2,2,3]]
=> [4,5,7,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,3],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,2],[2,3,3]]
=> [4,6,7,1,2,3,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,3],[2,2,3]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,3],[2,3,3]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,2,2],[2,2,3]]
=> [3,4,7,1,2,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,2,2],[2,3,3]]
=> [3,6,7,1,2,4,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,2,3],[2,2,3]]
=> [3,4,6,1,2,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,2,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,2,3],[2,3,3]]
=> [3,5,6,1,2,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,2,2,2],[2,3,3]]
=> [2,6,7,1,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,2,2,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,2,2,3],[2,3,3]]
=> [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,2,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[2,2,2,2],[3,3,3]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[2,2,2,3],[3,3,3]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1 - 1
[[1,1,1],[2,2],[3,3]]
=> [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 - 1
[[1,1,2],[2,2],[3,3]]
=> [6,7,3,4,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 - 1
[[1,1,3],[2,2],[3,3]]
=> [5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 - 1
[[1,1,1,1,2,2],[2,2]]
=> [5,6,1,2,3,4,7,8] => ?
=> ? => ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? => ? = 1 - 1
Description
The number of rafts of a permutation.
Let π be a permutation of length n. A small ascent of π is an index i such that π(i+1)=π(i)+1, see [[St000441]], and a raft of π is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St001186
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[2,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[2,3]]
=> [5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,3],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,3],[2,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[2,3]]
=> [4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,3],[2,3]]
=> [4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3,3],[2,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[2,3]]
=> [3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,3],[2,3]]
=> [3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3,3],[2,3]]
=> [3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3,3],[2,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,2],[2,3]]
=> [2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,3],[2,3]]
=> [2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3,3],[2,3]]
=> [2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,3,3],[2,3]]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [7,6,1,2,3,4,5] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,1,2],[2],[3]]
=> [7,5,1,2,3,4,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,1,3],[2],[3]]
=> [6,5,1,2,3,4,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,2,2],[2],[3]]
=> [7,4,1,2,3,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,2,3],[2],[3]]
=> [6,4,1,2,3,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,4,1,2,3,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
Description
Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001266
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001266: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001266: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[2,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[2,3]]
=> [5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,3],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,3],[2,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[2,3]]
=> [4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,3],[2,3]]
=> [4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3,3],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3,3],[2,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[2,3]]
=> [3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,3],[2,3]]
=> [3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3,3],[2,3]]
=> [3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3,3],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3,3],[2,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,2],[2,3]]
=> [2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,3],[2,3]]
=> [2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3,3],[2,3]]
=> [2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,3,3],[2,3]]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,2,2,2],[3,3]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,2,2,3],[3,3]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,2,3,3],[3,3]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2,2,3,3,3],[3,3]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[2],[3]]
=> [7,6,1,2,3,4,5] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,1,2],[2],[3]]
=> [7,5,1,2,3,4,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,1,3],[2],[3]]
=> [6,5,1,2,3,4,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,2,2],[2],[3]]
=> [7,4,1,2,3,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,2,3],[2],[3]]
=> [6,4,1,2,3,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1,1,3,3],[2],[3]]
=> [5,4,1,2,3,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
Description
The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra.
Matching statistic: St001195
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [[[.,.],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [[[.,.],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [[[.,.],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Matching statistic: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,1,2,4,3,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[3,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[4,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[5,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[3,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,3],[2,5]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,2],[4,5]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[[1,1],[2],[5]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[3],[5]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4],[5]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3],[5]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,5],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[4],[5]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001820The size of the image of the pop stack sorting operator. St001864The number of excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St000929The constant term of the character polynomial of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St001754The number of tolerances of a finite lattice. St000640The rank of the largest boolean interval in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000635The number of strictly order preserving maps of a poset into itself. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001631The number of simple modules S with dimExt1(S,A)=1 in the incidence algebra A 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. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. 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. St000782The indicator function of whether a given perfect matching is an L & P matching. St001207The 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]/(xn). 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. St001625The Möbius invariant of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000094The depth of an ordered tree. 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. 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. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. 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. St001171The vector space dimension of Ext1A(Io,A) when Io is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(xn). St000189The number of elements in the poset. 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. St000529The number of permutations whose descent word is the given binary word. St000180The number of chains 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]/(xn). St000100The number of linear extensions of a poset. 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.
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