Identifier
-
Mp00127:
Permutations
—left-to-right-maxima to Dyck path⟶
Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000914: Posets ⟶ ℤ
Values
[1,2] => [1,0,1,0] => [1,1,0,0] => ([(0,1)],2) => 1
[2,1] => [1,1,0,0] => [1,0,1,0] => ([(0,1)],2) => 1
[1,2,3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => ([(0,2),(2,1)],3) => 1
[1,3,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => ([(0,2),(2,1)],3) => 1
[2,1,3] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => ([(0,2),(2,1)],3) => 1
[2,3,1] => [1,1,0,1,0,0] => [1,1,1,0,0,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 1
[3,2,1] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => ([(0,3),(2,1),(3,2)],4) => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.
Map
parallelogram poset
Description
The cell poset of the parallelogram polyomino corresponding to the Dyck path.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
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