- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤ

Values

[1] => [1,0] => [1] => 1

[1,2] => [1,0,1,0] => [1,1] => 1

[2,1] => [1,1,0,0] => [2] => 1

[1,2,3] => [1,0,1,0,1,0] => [1,1,1] => 1

[1,3,2] => [1,0,1,1,0,0] => [1,2] => 2

[2,1,3] => [1,1,0,0,1,0] => [2,1] => 2

[2,3,1] => [1,1,0,1,0,0] => [3] => 1

[3,1,2] => [1,1,1,0,0,0] => [3] => 1

[3,2,1] => [1,1,1,0,0,0] => [3] => 1

[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => 1

[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,2] => 3

[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,2,1] => 5

[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,3] => 3

[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,3] => 3

[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,3] => 3

[2,1,3,4] => [1,1,0,0,1,0,1,0] => [2,1,1] => 3

[2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,2] => 5

[2,3,1,4] => [1,1,0,1,0,0,1,0] => [3,1] => 3

[2,3,4,1] => [1,1,0,1,0,1,0,0] => [4] => 1

[2,4,1,3] => [1,1,0,1,1,0,0,0] => [4] => 1

[2,4,3,1] => [1,1,0,1,1,0,0,0] => [4] => 1

[3,1,2,4] => [1,1,1,0,0,0,1,0] => [3,1] => 3

[3,1,4,2] => [1,1,1,0,0,1,0,0] => [4] => 1

[3,2,1,4] => [1,1,1,0,0,0,1,0] => [3,1] => 3

[3,2,4,1] => [1,1,1,0,0,1,0,0] => [4] => 1

[3,4,1,2] => [1,1,1,0,1,0,0,0] => [4] => 1

[3,4,2,1] => [1,1,1,0,1,0,0,0] => [4] => 1

[4,1,2,3] => [1,1,1,1,0,0,0,0] => [4] => 1

[4,1,3,2] => [1,1,1,1,0,0,0,0] => [4] => 1

[4,2,1,3] => [1,1,1,1,0,0,0,0] => [4] => 1

[4,2,3,1] => [1,1,1,1,0,0,0,0] => [4] => 1

[4,3,1,2] => [1,1,1,1,0,0,0,0] => [4] => 1

[4,3,2,1] => [1,1,1,1,0,0,0,0] => [4] => 1

[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => 1

[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => 4

[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => 9

[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => 6

[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 6

[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 6

[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => 9

[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,2,2] => 16

[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,3,1] => 11

[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,4] => 4

[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,4] => 4

[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,4] => 4

[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 11

[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,4] => 4

[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 11

[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,4] => 4

[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4] => 4

[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,4] => 4

[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 4

[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 4

[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 4

[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 4

[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 4

[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 4

[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => 4

[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [2,1,2] => 11

[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [2,2,1] => 16

[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [2,3] => 9

[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => 9

[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => 9

[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [3,1,1] => 6

[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [3,2] => 9

[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [4,1] => 4

[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [5] => 1

[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [5] => 1

[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [5] => 1

[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [4,1] => 4

[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [5] => 1

[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [4,1] => 4

[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [5] => 1

[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [5] => 1

[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [5] => 1

[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [5] => 1

[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [5] => 1

[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [5] => 1

[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [5] => 1

[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [5] => 1

[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [5] => 1

[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 6

[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => 9

[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [4,1] => 4

[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [5] => 1

[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [5] => 1

[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [5] => 1

[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 6

[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => 9

[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [4,1] => 4

[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [5] => 1

[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [5] => 1

[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [5] => 1

[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 4

[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [5] => 1

[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 4

[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [5] => 1

[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [5] => 1

[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [5] => 1

[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [5] => 1

[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [5] => 1

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Description

The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

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.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.