Identifier
Values
[1] => [1,0,1,0] => [1,1,0,0] => 2
[2] => [1,1,0,0,1,0] => [1,0,1,0,1,0] => 4
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
[2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => 3
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => 5
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 4
[3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0] => 4
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => 3
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 5
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 5
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 4
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => 3
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 5
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 3
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 2
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 4
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 4
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => 3
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => 3
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 3
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 4
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 5
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 2
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 3
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 4
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 3
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => 4
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 3
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 6
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 3
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 3
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => 4
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 2
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 3
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 3
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 4
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 5
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 4
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 5
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 3
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 3
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 4
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 4
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 4
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 3
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 6
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 4
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 2
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 3
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 6
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 3
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 4
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 6
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 3
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 5
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 4
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 5
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 3
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 4
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 3
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 4
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 4
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 3
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2
>>> Load all 131 entries. <<<
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 3
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 3
[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 3
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 4
[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 4
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 7
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 3
[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 4
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 3
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 3
[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 6
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 3
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 5
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 4
[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3
[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 3
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 3
[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 2
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Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
Map
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.