Identifier
Values
{{1,2,3}} => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 2
{{1},{2},{3}} => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 2
{{1,2,3,4}} => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 3
{{1},{2},{3},{4}} => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 3
{{1,2,3,4,5}} => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
{{1,2,3,4,5,6}} => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
{{1},{2},{3},{4},{5},{6}} => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
{{1,2,3,4,5,6,7}} => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
{{1},{2},{3},{4},{5},{6},{7}} => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
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Description
The 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.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.