Your data matches 98 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: 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]
 => 3
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000297
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => 100 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => 100 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => 100 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => 100 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => 110 => 2
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => 100 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 110 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 3
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => 100 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => 100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000667
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: 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]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 3
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000733
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ 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],[2]]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001264
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001264: 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]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
Description
The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
Matching statistic: St001265
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001265: 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]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
Description
The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001803
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ 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],[2]]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●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]
 => 3
[[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
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,2],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,3],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,4],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[3,3],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[3,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,3],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,4],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[3,3],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[2,4],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[3,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[2,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[3,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[2,2],[3,3],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[2,2],[3,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[2,3],[3,4],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1,1],[2,2],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1,1],[2,3],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1,2],[2,2],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1,2],[2,3],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1,3],[2,2],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1,3],[2,3],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2,2],[2,3],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2,3],[2,3],[3]]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,2],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,3],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[2,5],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[3,3],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[3,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[3,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[3,5],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[4,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,1],[4,5],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,3],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[2,5],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[3,3],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1],[2],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[2],[3],[4]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1],[2,3]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1],[3,3]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[2,3]]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3,3]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,2],[3,3]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,3],[2],[3]]
 => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1],[2],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[2],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[3],[4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[1,1],[2,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1],[3,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1],[4,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[2,4]]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2,4]]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[4,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[3,4]]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[4,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,2],[3,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,2],[4,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,3],[3,4]]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2,3],[4,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[3,3],[4,4]]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,1],[3],[4]]
 => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,4],[2],[4]]
 => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,3],[3],[4]]
 => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,4],[3],[4]]
 => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[2,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[2,3],[3],[4]]
 => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[2,4],[3],[4]]
 => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,2],[4]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,3],[4]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,4],[3]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,4],[4]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[3,3],[4]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[3,4],[4]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,3],[4]]
 => [5,2,4,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,4],[3]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,4],[4]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[3,3],[4]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,3],[2,4],[3]]
 => [3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[3,4],[4]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,3],[2,4],[4]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,3],[3,4],[4]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[2,2],[3,3],[4]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[2,2],[3,4],[4]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[2,3],[3,4],[4]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1,1],[2,2],[3]]
 => [6,4,5,1,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,1,1],[2,3],[3]]
 => [5,4,6,1,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,1,2],[2,2],[3]]
 => [6,3,4,1,2,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,1,2],[2,3],[3]]
 => [5,3,6,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,1,3],[2,2],[3]]
 => [5,3,4,1,2,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,1,3],[2,3],[3]]
 => [4,3,5,1,2,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,2,2],[2,3],[3]]
 => [5,2,6,1,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,2,3],[2,3],[3]]
 => [4,2,5,1,3,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[[1,1],[2,2],[3,3]]
 => [5,6,3,4,1,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[[1,1,1,1,1],[2,2]]
 => [6,7,1,2,3,4,5] => [2,5] => ([(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,1,1,1,2],[2,2]]
 => [5,6,1,2,3,4,7] => [2,5] => ([(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,1,1,2,2],[2,2]]
 => [4,5,1,2,3,6,7] => [2,5] => ([(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,1,2,2,2],[2,2]]
 => [3,4,1,2,5,6,7] => [2,5] => ([(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,1,1,1],[2,2,2]]
 => [5,6,7,1,2,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,1,1,2],[2,2,2]]
 => [4,5,6,1,2,3,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[[1,1],[2,2],[5]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,3],[5]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,5],[3]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,4],[5]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,5],[4]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[2,5],[5]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[3,3],[5]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[3,5],[5]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[4,4],[5]]
 => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,1],[4,5],[5]]
 => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,3],[5]]
 => [5,2,4,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,5],[3]]
 => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[[1,2],[2,4],[5]]
 => [5,2,4,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
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.
Matching statistic: St001553
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 80%
Values
[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[1,1],[2,3]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,1],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,2],[2,3]]
 => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[[1,2],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[2,2],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[1,1],[2,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,1],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,1],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,2],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[[1,2],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[[1,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[2,2],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[2,2],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[2,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[[2,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[3,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[[1,1],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[[2,2],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1],[2,3]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1],[3,3]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2],[2,3]]
 => [4,6,1,2,3,5] => [6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,3],[2,2]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,1,2],[3,3]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,3],[2,3]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,1,3],[3,3]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,2,2],[2,3]]
 => [3,6,1,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,3],[2,2]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,3],[2,3]]
 => [3,5,1,2,4,6] => [5,1,3,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,3,3],[2,2]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,1,2,3],[3,3]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,3,3],[2,3]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,1,3,3],[3,3]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,2,2,2],[2,3]]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,2],[3,3]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,3],[2,3]]
 => [2,5,1,3,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,2,2,3],[3,3]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,2,3,3],[2,3]]
 => [2,4,1,3,5,6] => [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,2,3,3],[3,3]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[2,2,2,2],[3,3]]
 => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[2,2,2,3],[3,3]]
 => [4,5,1,2,3,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[[2,2,3,3],[3,3]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[[1,1,1,1],[2],[3]]
 => [6,5,1,2,3,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[[1,1,1,2],[2],[3]]
 => [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 2
[[1,1,1,3],[2],[3]]
 => [5,4,1,2,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 2
[[1,1,2,2],[2],[3]]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 2
[[1,1,2,3],[2],[3]]
 => [5,3,1,2,4,6] => [3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 2
[[1,1,3,3],[2],[3]]
 => [4,3,1,2,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 2
[[1,2,2,2],[2],[3]]
 => [6,2,1,3,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[[1,2,2,3],[2],[3]]
 => [5,2,1,3,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 2
[[1,2,3,3],[2],[3]]
 => [4,2,1,3,5,6] => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 2
[[1,3,3,3],[2],[3]]
 => [3,2,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[[1,1,1],[2,2,3]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1],[2,3,3]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1],[3,3,3]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2],[2,2,3]]
 => [3,4,6,1,2,5] => [6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2],[2,3,3]]
 => [3,5,6,1,2,4] => [6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2],[3,3,3]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2],[2,3,3]]
 => [2,5,6,1,3,4] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2],[3,3,3]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[2,2,2],[3,3,3]]
 => [4,5,6,1,2,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1],[2,2],[3]]
 => [6,4,5,1,2,3] => [5,1,2,6,4,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 1
[[1,1,1],[2,3],[3]]
 => [5,4,6,1,2,3] => [6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2],[2,2],[3]]
 => [6,3,4,1,2,5] => [4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.
The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001866The nesting alignments of a signed permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000911The number of maximal antichains of maximal size in a poset. St001820The size of the image of the pop stack sorting operator. 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. 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. 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. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000080The rank of the poset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. 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. 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. St001890The maximum magnitude of the Möbius function of a poset. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. 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. St001964The interval resolution global dimension of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000189The number of elements in the poset. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000307The number of rowmotion orbits of a poset. 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 $K[x]/(x^n)$. 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. St000422The energy of a graph, if it is integral.