Identifier
-
Mp00102:
Dyck paths
—rise composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001268: Posets ⟶ ℤ
Values
[1,0] => [1] => [[1],[]] => ([],1) => 1
[1,0,1,0] => [1,1] => [[1,1],[]] => ([(0,1)],2) => 1
[1,1,0,0] => [2] => [[2],[]] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 1
[1,0,1,1,0,0] => [1,2] => [[2,1],[]] => ([(0,1),(0,2)],3) => 2
[1,1,0,0,1,0] => [2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3) => 2
[1,1,0,1,0,0] => [2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3) => 2
[1,1,1,0,0,0] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,1,0,0] => [1,1,2] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 3
[1,0,1,1,0,0,1,0] => [1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 4
[1,0,1,1,0,1,0,0] => [1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 4
[1,0,1,1,1,0,0,0] => [1,3] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 3
[1,1,0,0,1,0,1,0] => [2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,0,0,1,1,0,0] => [2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 4
[1,1,0,1,0,0,1,0] => [2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,0,1,0,1,0,0] => [2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,0,1,1,0,0,0] => [2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 4
[1,1,1,0,0,0,1,0] => [3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,1,0,0,1,0,0] => [3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,1,0,1,0,0,0] => [3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 3
[1,1,1,1,0,0,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 4
[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 5
[1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 5
[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 4
[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 5
[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5) => 5
[1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 5
[1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 5
[1,0,1,1,0,1,1,0,0,0] => [1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5) => 5
[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5
[1,0,1,1,1,0,0,1,0,0] => [1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5
[1,0,1,1,1,0,1,0,0,0] => [1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5
[1,0,1,1,1,1,0,0,0,0] => [1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 4
[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5
[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 5
[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 5
[1,1,0,0,1,1,1,0,0,0] => [2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 5
[1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,0,1,0,0,1,1,0,0] => [2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5
[1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,0,1,0,1,1,0,0,0] => [2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5
[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 5
[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 5
[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 5
[1,1,0,1,1,1,0,0,0,0] => [2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 5
[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 4
[1,1,1,0,0,0,1,1,0,0] => [3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 5
[1,1,1,0,0,1,0,0,1,0] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 4
[1,1,1,0,0,1,0,1,0,0] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 4
[1,1,1,0,0,1,1,0,0,0] => [3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 5
[1,1,1,0,1,0,0,0,1,0] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 4
[1,1,1,0,1,0,0,1,0,0] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 4
[1,1,1,0,1,0,1,0,0,0] => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 4
[1,1,1,0,1,1,0,0,0,0] => [3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 5
[1,1,1,1,0,0,0,0,1,0] => [4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,1,1,0,0,1,0,0,0] => [4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,1,1,0,1,0,0,0,0] => [4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[1,1,1,1,1,0,0,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 5
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 6
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 6
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 5
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 6
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 6
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 6
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 6
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Description
The size of the largest ordinal summand in the poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
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.
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
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
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.
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.
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