Identifier
-
Mp00113:
Perfect matchings
—reverse⟶
Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000028: Permutations ⟶ ℤ
Values
[(1,2)] => [(1,2)] => [2,1] => [1] => 0
[(1,2),(3,4)] => [(1,2),(3,4)] => [2,1,4,3] => [2,1,3] => 1
[(1,3),(2,4)] => [(1,3),(2,4)] => [3,4,1,2] => [3,1,2] => 1
[(1,4),(2,3)] => [(1,4),(2,3)] => [3,4,2,1] => [3,2,1] => 1
[(1,2),(3,4),(5,6)] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,5] => 1
[(1,3),(2,4),(5,6)] => [(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,4] => 1
[(1,4),(2,3),(5,6)] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,5,4,3] => 1
[(1,5),(2,3),(4,6)] => [(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,5,1,4,2] => 2
[(1,6),(2,3),(4,5)] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [3,5,2,4,1] => 2
[(1,6),(2,4),(3,5)] => [(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [4,5,2,3,1] => 2
[(1,5),(2,4),(3,6)] => [(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,5,1,3,2] => 2
[(1,4),(2,5),(3,6)] => [(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,5,1,2,3] => 2
[(1,3),(2,5),(4,6)] => [(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,5,1,2,4] => 2
[(1,2),(3,5),(4,6)] => [(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,4,1,2,5] => 2
[(1,2),(3,6),(4,5)] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [3,4,2,1,5] => 2
[(1,3),(2,6),(4,5)] => [(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [3,5,2,1,4] => 2
[(1,4),(2,6),(3,5)] => [(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [4,5,2,1,3] => 2
[(1,5),(2,6),(3,4)] => [(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [4,5,3,1,2] => 2
[(1,6),(2,5),(3,4)] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,5,3,2,1] => 2
[(1,5),(2,6),(3,7),(4,8)] => [(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => 3
[(1,2),(3,6),(4,7),(5,8)] => [(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,7] => 3
[(1,2),(3,8),(4,7),(5,6)] => [(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [4,5,6,3,2,1,7] => 3
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => 4
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Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ (OEIS:A000108). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ (OEIS:A000139).
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ (OEIS:A000108). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ (OEIS:A000139).
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
This map is undefined for the empty permutation.
Map
reverse
Description
The reverse of a perfect matching of $\{1,2,...,n\}$.
This is the perfect matching obtained by replacing $i$ by $n+1-i$.
This is the perfect matching obtained by replacing $i$ by $n+1-i$.
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