- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000003: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,0,1,0] => [3,1,1] => 6

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 768

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 168

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 216

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => 70

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 14

[1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 216

[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 56

[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => 90

[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 35

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 9

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 20

[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 10

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

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

[1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 168

[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 42

[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 56

[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 21

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

[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => 70

[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 21

[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 35

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

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

[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 10

[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 6

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

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

[1,1,1,0,0,0,1,0,1,0] => [4,3] => 14

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 292864

[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 48048

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 64064

[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => 15015

[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 2112

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 68640

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 12870

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => 21450

[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => 5775

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => 990

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 3520

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => 1155

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => 288

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 42

[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 64064

[1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 11583

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 16016

[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => 4158

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 660

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => 21450

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => 4455

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => 7700

[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => 2310

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => 450

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => 1540

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => 567

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => 162

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => 28

[1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 3520

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 825

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => 1540

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => 525

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => 120

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => 448

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => 189

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => 64

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => 14

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + q^{2}$

$F_{4} = 6\ q + 2\ q^{2} + 2\ q^{3} + 2\ q^{5} + q^{6} + q^{16}$

$F_{5} = 8\ q + 2\ q^{2} + 2\ q^{3} + 2\ q^{4} + 4\ q^{5} + q^{6} + 2\ q^{9} + 2\ q^{10} + 2\ q^{14} + q^{16} + q^{20} + 2\ q^{21} + 2\ q^{35} + q^{42} + 2\ q^{56} + 2\ q^{70} + q^{90} + 2\ q^{168} + 2\ q^{216} + q^{768}$

$F_{6} = 10\ q + 2\ q^{2} + 2\ q^{3} + 2\ q^{4} + 6\ q^{5} + q^{6} + 2\ q^{9} + 2\ q^{10} + 6\ q^{14} + 2\ q^{15} + q^{16} + q^{20} + 2\ q^{21} + 2\ q^{28} + 4\ q^{35} + 4\ q^{42} + 2\ q^{56} + 2\ q^{64} + 3\ q^{70} + 2\ q^{84} + q^{90} + 2\ q^{120} + 2\ q^{162} + 2\ q^{168} + 2\ q^{189} + 2\ q^{210} + 2\ q^{216} + 2\ q^{252} + 2\ q^{288} + 2\ q^{300} + q^{448} + 2\ q^{450} + 2\ q^{462} + 2\ q^{525} + 2\ q^{567} + 2\ q^{660} + q^{768} + 2\ q^{825} + 2\ q^{990} + 2\ q^{1155} + q^{1188} + 2\ q^{1320} + 2\ q^{1540} + 2\ q^{2112} + 2\ q^{2310} + q^{2640} + 2\ q^{2970} + 2\ q^{3520} + 2\ q^{4158} + 2\ q^{4455} + 2\ q^{5775} + q^{7700} + q^{8580} + 2\ q^{11583} + 2\ q^{12870} + 2\ q^{15015} + q^{16016} + 2\ q^{21450} + 2\ q^{48048} + 2\ q^{64064} + q^{68640} + q^{292864}$

Description

The number of standard Young tableaux of the partition.

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition

$\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition

$(n-1,\ldots,2,1)$ when cutting out

$D$ considered as a path from

$(0,0)$ to

$(n,n)$ .
In other words,

$\lambda_{i}$ is the number of down-steps before the

$(n+1-i)$ -th up-step of

$D$ .
This map is a bijection between Dyck paths of size

$n$ and partitions inside the staircase partition

$(n-1,\ldots,2,1)$ .