- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001097: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => [3,2,1] => 10

[1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [3,2,1] => 10

[1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => [2,2,1] => 12

[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [2,2,1] => 12

[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [2,2,1] => 12

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

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

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

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

[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [2,1,1] => 6

[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [1,1,1] => 6

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

[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [1,1,1] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [4,3,2,1] => 1074

[1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => [4,3,2,1] => 1074

[1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => [3,3,2,1] => 836

[1,0,1,0,1,0,1,1,0,1,0,0] => [4,3,3,2,1] => [3,3,2,1] => 836

[1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => [3,3,2,1] => 836

[1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => [4,2,2,1] => 336

[1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => [4,2,2,1] => 336

[1,0,1,0,1,1,0,1,0,0,1,0] => [5,3,2,2,1] => [3,2,2,1] => 384

[1,0,1,0,1,1,0,1,0,1,0,0] => [4,3,2,2,1] => [3,2,2,1] => 384

[1,0,1,0,1,1,0,1,1,0,0,0] => [3,3,2,2,1] => [3,2,2,1] => 384

[1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => [2,2,2,1] => 246

[1,0,1,0,1,1,1,0,0,1,0,0] => [4,2,2,2,1] => [2,2,2,1] => 246

[1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [2,2,2,1] => 246

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [2,2,2,1] => 246

[1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => [4,3,1,1] => 186

[1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => [4,3,1,1] => 186

[1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => [3,3,1,1] => 184

[1,0,1,1,0,0,1,1,0,1,0,0] => [4,3,3,1,1] => [3,3,1,1] => 184

[1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => [3,3,1,1] => 184

[1,0,1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,1] => [4,2,1,1] => 54

[1,0,1,1,0,1,0,0,1,1,0,0] => [4,4,2,1,1] => [4,2,1,1] => 54

[1,0,1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,1] => [3,2,1,1] => 96

[1,0,1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,1] => [3,2,1,1] => 96

[1,0,1,1,0,1,0,1,1,0,0,0] => [3,3,2,1,1] => [3,2,1,1] => 96

[1,0,1,1,0,1,1,0,0,0,1,0] => [5,2,2,1,1] => [2,2,1,1] => 84

[1,0,1,1,0,1,1,0,0,1,0,0] => [4,2,2,1,1] => [2,2,1,1] => 84

[1,0,1,1,0,1,1,0,1,0,0,0] => [3,2,2,1,1] => [2,2,1,1] => 84

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [2,2,1,1] => 84

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

[1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => [4,1,1,1] => 6

[1,0,1,1,1,0,0,1,0,0,1,0] => [5,3,1,1,1] => [3,1,1,1] => 24

[1,0,1,1,1,0,0,1,0,1,0,0] => [4,3,1,1,1] => [3,1,1,1] => 24

[1,0,1,1,1,0,0,1,1,0,0,0] => [3,3,1,1,1] => [3,1,1,1] => 24

[1,0,1,1,1,0,1,0,0,0,1,0] => [5,2,1,1,1] => [2,1,1,1] => 36

[1,0,1,1,1,0,1,0,0,1,0,0] => [4,2,1,1,1] => [2,1,1,1] => 36

[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [2,1,1,1] => 36

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [2,1,1,1] => 36

[1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => [1,1,1,1] => 24

[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [1,1,1,1] => 24

[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [1,1,1,1] => 24

[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [1,1,1,1] => 24

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [1,1,1,1] => 24

[1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => [4,3,2] => 79

[1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => [4,3,2] => 79

[1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => [3,3,2] => 74

[1,1,0,0,1,0,1,1,0,1,0,0] => [4,3,3,2] => [3,3,2] => 74

[1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => [3,3,2] => 74

[1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => [4,2,2] => 24

[1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => [4,2,2] => 24

[1,1,0,0,1,1,0,1,0,0,1,0] => [5,3,2,2] => [3,2,2] => 38

[1,1,0,0,1,1,0,1,0,1,0,0] => [4,3,2,2] => [3,2,2] => 38

[1,1,0,0,1,1,0,1,1,0,0,0] => [3,3,2,2] => [3,2,2] => 38

[1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => [2,2,2] => 30

[1,1,0,0,1,1,1,0,0,1,0,0] => [4,2,2,2] => [2,2,2] => 30

[1,1,0,0,1,1,1,0,1,0,0,0] => [3,2,2,2] => [2,2,2] => 30

[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [2,2,2] => 30

[1,1,0,1,0,0,1,0,1,0,1,0] => [5,4,3,1] => [4,3,1] => 14

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

[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,3,1] => [3,3,1] => 18

[1,1,0,1,0,0,1,1,0,1,0,0] => [4,3,3,1] => [3,3,1] => 18

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3,3,1] => [3,3,1] => 18

>>> Load all 1700 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ in the formal group law for linear orders, with generating function $f(x) = x/(1-x)$, see [1, sec. 3.4].
This statistic gives the number of Smirnov arrangements of a set of letters with $\lambda_i$ of the $i$th letter, where a Smirnov word is a word with no repeated adjacent letters. e.g., [3,2,1] = > 10 since there are 10 Smirnov rearrangements of the word 'aaabbc': 'ababac', 'ababca', 'abacab', 'abacba', 'abcaba', 'acabab', 'acbaba', 'babaca', 'bacaba', 'cababa'.

Map

first row removal

Description

Removes the first entry of an integer 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 $(n-1,\ldots,2,1)$.