- FindStat

Identifier

Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000548: Integer partitions ⟶ ℤ

Values

=> ([(0,1)],2) => [2] => [1,1] => 2
=> ([(0,1)],2) => [2] => [1,1] => 2
=> ([(0,2),(2,1)],3) => [3] => [1,1,1] => 3
=> ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 4
=> ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 4
=> ([(0,2),(2,1)],3) => [3] => [1,1,1] => 3
=> ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1] => 4
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 6
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2] => [2,2,1,1] => 6
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 6
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 6
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2] => [2,2,1,1] => 6
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 6
=> ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1] => 4
=> ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1] => 5
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 8
=> ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => [5,3] => [2,2,2,1,1] => 8
=> ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 8
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 8
=> ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8) => [5,3] => [2,2,2,1,1] => 8
=> ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 9
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 8
=> ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1] => 5
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1] => 6
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => [2,2,2,2,1,1] => 10
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10) => [6,4] => [2,2,2,2,1,1] => 10
=> ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => [2,2,2,2,1,1] => 10
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => [2,2,2,2,1,1] => 10
=> ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10) => [6,4] => [2,2,2,2,1,1] => 10
=> ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12) => [6,4,2] => [3,3,2,2,1,1] => 12
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => [2,2,2,2,1,1] => 10
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1] => 6
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [7] => [1,1,1,1,1,1,1] => 7
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12) => [7,5] => [2,2,2,2,2,1,1] => 12
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12) => [7,5] => [2,2,2,2,2,1,1] => 12
011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12) => [7,5] => [2,2,2,2,2,1,1] => 12
100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12) => [7,5] => [2,2,2,2,2,1,1] => 12
101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12) => [7,5] => [2,2,2,2,2,1,1] => 12
111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12) => [7,5] => [2,2,2,2,2,1,1] => 12
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [7] => [1,1,1,1,1,1,1] => 7
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => [1,1,1,1,1,1,1,1] => 8
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => [1,1,1,1,1,1,1,1] => 8
00000000 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => [9] => [1,1,1,1,1,1,1,1,1] => 9
11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => [9] => [1,1,1,1,1,1,1,1,1] => 9
000000000 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [10] => [1,1,1,1,1,1,1,1,1,1] => 10
111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [10] => [1,1,1,1,1,1,1,1,1,1] => 10

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$F_{1} = 2\ q^{2}$

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

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

$F_{4} = 2\ q^{5} + 6\ q^{8} + 8\ q^{9}$

Description

The number of different non-empty partial sums of an integer partition.

Map

Greene-Kleitman invariant

Description

The Greene-Kleitman invariant of a poset.
This is the partition

$(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$ , where

$c_k$ is the maximum cardinality of a union of

$k$ chains of the poset. Equivalently, this is the conjugate of the partition

$(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$ , where

$a_k$ is the maximum cardinality of a union of

$k$ antichains of the poset.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

Map

poset of factors

Description

The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that

$u < v$ if and only if

$u$ is a factor of

$v$ .