- FindStat

Identifier

Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000492: Set partitions ⟶ ℤ

Values

{{1,2}} => {{1,2}} => 0

{{1},{2}} => {{1},{2}} => 1

{{1,2,3}} => {{1,2,3}} => 0

{{1,2},{3}} => {{1},{2,3}} => 1

{{1,3},{2}} => {{1,2},{3}} => 2

{{1},{2,3}} => {{1,3},{2}} => 1

{{1},{2},{3}} => {{1},{2},{3}} => 3

{{1,2,3,4}} => {{1,2,3,4}} => 0

{{1,2,3},{4}} => {{1},{2,3,4}} => 1

{{1,2,4},{3}} => {{1,2,3},{4}} => 3

{{1,2},{3,4}} => {{1,4},{2,3}} => 1

{{1,2},{3},{4}} => {{1},{2,3},{4}} => 4

{{1,3,4},{2}} => {{1,2,4},{3}} => 2

{{1,3},{2,4}} => {{1,3},{2,4}} => 1

{{1,3},{2},{4}} => {{1},{2,4},{3}} => 3

{{1,4},{2,3}} => {{1,2},{3,4}} => 2

{{1},{2,3,4}} => {{1,3,4},{2}} => 1

{{1},{2,3},{4}} => {{1},{2},{3,4}} => 3

{{1,4},{2},{3}} => {{1,2},{3},{4}} => 5

{{1},{2,4},{3}} => {{1,3},{2},{4}} => 4

{{1},{2},{3,4}} => {{1,4},{2},{3}} => 3

{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => 6

{{1,2,3,4,5}} => {{1,2,3,4,5}} => 0

{{1,2,3,4},{5}} => {{1},{2,3,4,5}} => 1

{{1,2,3,5},{4}} => {{1,2,3,4},{5}} => 4

{{1,2,3},{4,5}} => {{1,5},{2,3,4}} => 1

{{1,2,3},{4},{5}} => {{1},{2,3,4},{5}} => 5

{{1,2,4,5},{3}} => {{1,2,3,5},{4}} => 3

{{1,2,4},{3,5}} => {{1,4},{2,3,5}} => 1

{{1,2,4},{3},{5}} => {{1},{2,3,5},{4}} => 4

{{1,2,5},{3,4}} => {{1,2,3},{4,5}} => 3

{{1,2},{3,4,5}} => {{1,4,5},{2,3}} => 1

{{1,2},{3,4},{5}} => {{1},{2,3},{4,5}} => 4

{{1,2,5},{3},{4}} => {{1,2,3},{4},{5}} => 7

{{1,2},{3,5},{4}} => {{1,4},{2,3},{5}} => 5

{{1,2},{3},{4,5}} => {{1,5},{2,3},{4}} => 4

{{1,2},{3},{4},{5}} => {{1},{2,3},{4},{5}} => 8

{{1,3,4,5},{2}} => {{1,2,4,5},{3}} => 2

{{1,3,4},{2,5}} => {{1,3},{2,4,5}} => 1

{{1,3,4},{2},{5}} => {{1},{2,4,5},{3}} => 3

{{1,3,5},{2,4}} => {{1,2,4},{3,5}} => 2

{{1,3},{2,4,5}} => {{1,3,5},{2,4}} => 1

{{1,3},{2,4},{5}} => {{1},{2,4},{3,5}} => 3

{{1,3,5},{2},{4}} => {{1,2,4},{3},{5}} => 6

{{1,3},{2,5},{4}} => {{1,3},{2,4},{5}} => 5

{{1,3},{2},{4,5}} => {{1,5},{2,4},{3}} => 3

{{1,3},{2},{4},{5}} => {{1},{2,4},{3},{5}} => 7

{{1,4,5},{2,3}} => {{1,2,5},{3,4}} => 2

{{1,4},{2,3,5}} => {{1,3,4},{2,5}} => 1

{{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => 3

{{1,5},{2,3,4}} => {{1,2},{3,4,5}} => 2

{{1},{2,3,4,5}} => {{1,3,4,5},{2}} => 1

{{1},{2,3,4},{5}} => {{1},{2},{3,4,5}} => 3

{{1,5},{2,3},{4}} => {{1,2},{3,4},{5}} => 6

{{1},{2,3,5},{4}} => {{1,3,4},{2},{5}} => 5

{{1},{2,3},{4,5}} => {{1,5},{2},{3,4}} => 3

{{1},{2,3},{4},{5}} => {{1},{2},{3,4},{5}} => 7

{{1,4,5},{2},{3}} => {{1,2,5},{3},{4}} => 5

{{1,4},{2,5},{3}} => {{1,3},{2,5},{4}} => 4

{{1,4},{2},{3,5}} => {{1,4},{2,5},{3}} => 3

{{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => 6

{{1,5},{2,4},{3}} => {{1,2},{3,5},{4}} => 5

{{1},{2,4,5},{3}} => {{1,3,5},{2},{4}} => 4

{{1},{2,4},{3,5}} => {{1,4},{2},{3,5}} => 3

{{1},{2,4},{3},{5}} => {{1},{2},{3,5},{4}} => 6

{{1,5},{2},{3,4}} => {{1,2},{3},{4,5}} => 5

{{1},{2,5},{3,4}} => {{1,3},{2},{4,5}} => 4

{{1},{2},{3,4,5}} => {{1,4,5},{2},{3}} => 3

{{1},{2},{3,4},{5}} => {{1},{2},{3},{4,5}} => 6

{{1,5},{2},{3},{4}} => {{1,2},{3},{4},{5}} => 9

{{1},{2,5},{3},{4}} => {{1,3},{2},{4},{5}} => 8

{{1},{2},{3,5},{4}} => {{1,4},{2},{3},{5}} => 7

{{1},{2},{3},{4,5}} => {{1,5},{2},{3},{4}} => 6

{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => 10

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 0

{{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => 1

{{1,2,3,4,6},{5}} => {{1,2,3,4,5},{6}} => 5

{{1,2,3,4},{5,6}} => {{1,6},{2,3,4,5}} => 1

{{1,2,3,4},{5},{6}} => {{1},{2,3,4,5},{6}} => 6

{{1,2,3,5,6},{4}} => {{1,2,3,4,6},{5}} => 4

{{1,2,3,5},{4,6}} => {{1,5},{2,3,4,6}} => 1

{{1,2,3,5},{4},{6}} => {{1},{2,3,4,6},{5}} => 5

{{1,2,3,6},{4,5}} => {{1,2,3,4},{5,6}} => 4

{{1,2,3},{4,5,6}} => {{1,5,6},{2,3,4}} => 1

{{1,2,3},{4,5},{6}} => {{1},{2,3,4},{5,6}} => 5

{{1,2,3,6},{4},{5}} => {{1,2,3,4},{5},{6}} => 9

{{1,2,3},{4,6},{5}} => {{1,5},{2,3,4},{6}} => 6

{{1,2,3},{4},{5,6}} => {{1,6},{2,3,4},{5}} => 5

{{1,2,3},{4},{5},{6}} => {{1},{2,3,4},{5},{6}} => 10

{{1,2,4,5,6},{3}} => {{1,2,3,5,6},{4}} => 3

{{1,2,4,5},{3,6}} => {{1,4},{2,3,5,6}} => 1

{{1,2,4,5},{3},{6}} => {{1},{2,3,5,6},{4}} => 4

{{1,2,4,6},{3,5}} => {{1,2,3,5},{4,6}} => 3

{{1,2,4},{3,5,6}} => {{1,4,6},{2,3,5}} => 1

{{1,2,4},{3,5},{6}} => {{1},{2,3,5},{4,6}} => 4

{{1,2,4,6},{3},{5}} => {{1,2,3,5},{4},{6}} => 8

{{1,2,4},{3,6},{5}} => {{1,4},{2,3,5},{6}} => 6

{{1,2,4},{3},{5,6}} => {{1,6},{2,3,5},{4}} => 4

{{1,2,4},{3},{5},{6}} => {{1},{2,3,5},{4},{6}} => 9

{{1,2,5,6},{3,4}} => {{1,2,3,6},{4,5}} => 3

{{1,2,5},{3,4,6}} => {{1,4,5},{2,3,6}} => 1

>>> Load all 1200 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2} + q^{3}$

$F_{4} = 1 + 4\ q + 2\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + 8\ q + 4\ q^{2} + 11\ q^{3} + 7\ q^{4} + 7\ q^{5} + 6\ q^{6} + 4\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + 16\ q + 8\ q^{2} + 31\ q^{3} + 20\ q^{4} + 22\ q^{5} + 30\ q^{6} + 19\ q^{7} + 19\ q^{8} + 11\ q^{9} + 12\ q^{10} + 6\ q^{11} + 4\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + 32\ q + 16\ q^{2} + 89\ q^{3} + 58\ q^{4} + 65\ q^{5} + 107\ q^{6} + 85\ q^{7} + 82\ q^{8} + 74\ q^{9} + 73\ q^{10} + 50\ q^{11} + 49\ q^{12} + 29\ q^{13} + 24\ q^{14} + 17\ q^{15} + 12\ q^{16} + 6\ q^{17} + 4\ q^{18} + 2\ q^{19} + q^{20} + q^{21}$

Description

The rob statistic of a set partition.
Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ .
According to [1, Definition 3], a rob (right-opener-bigger) of

$S$ is given by a pair

$i < j$ such that

$j = \operatorname{min} B_b$ and

$i \in B_a$ for

$a < b$ .
This is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.

Map

rotate increasing

Description

The rotation of the set partition obtained by adding 1 to each entry different from the largest, and replacing the largest entry with 1.