- FindStat

Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000491
(load all 3 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions 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], see also [2,3], an inversion 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 statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.

Matching statistic: St000496
(load all 3 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rcs 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 '''rcs''' (right-closer-smaller) of

$S$ is given by a pair

$i > j$ such that

$j = \operatorname{max} B_b$ and

$i \in B_a$ for

$a < b$ .

Matching statistic: St000581
(load all 3 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.

Matching statistic: St000497

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%

Values

[1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
[1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => {{1},{2,3}}
 => 0
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => 0
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 0
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 2
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 2
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1},{2,3,4}}
 => 0
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 1
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1,3,4},{2}}
 => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,3},{2},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => {{1},{2},{3,4},{5}}
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,4},{2},{3},{5}}
 => {{1,4},{2},{3},{5}}
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => {{1},{2},{3},{4,5}}
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1},{2,3},{4,5}}
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => {{1},{2},{3,5},{4}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => {{1},{2,5},{3},{4}}
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,4},{2,5},{3}}
 => {{1,5},{2,4},{3}}
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1},{2,4},{3,5}}
 => {{1},{2,5},{3,4}}
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,5},{2,3},{4}}
 => {{1,3},{2,5},{4}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1,4},{2},{3,5}}
 => {{1,5},{2},{3,4}}
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,3,5},{2,4}}
 => {{1,5},{2,3,4}}
 => 3
[1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => {{1},{2},{3,4,5}}
 => 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2},{3,4,5,6,7,8}}
 => {{1,2,3,4,5,6},{7},{8}}
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 0
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,4,5,6,7,8},{3}}
 => {{1,3,5,7},{2,4,6},{8}}
 => {{1,7},{2,3,4,5,6},{8}}
 => ? = 5
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,5,6,7,8},{4}}
 => {{1,3,5},{2,4,6,7},{8}}
 => {{1,6,7},{2,3,4,5},{8}}
 => ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,6,7,8},{5}}
 => {{1,3,5,6,7},{2,4},{8}}
 => {{1,5,6,7},{2,3,4},{8}}
 => ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,7,8},{6}}
 => {{1,3},{2,4,5,6,7},{8}}
 => {{1,4,5,6,7},{2,3},{8}}
 => ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6,7},{2},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => ? = 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,4,5,6,7,8},{2},{3}}
 => {{1,4,7},{2,5,8},{3,6}}
 => {{1,8},{2,7},{3,4,5,6}}
 => ? = 10
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,3,5,6,7,8},{2},{4}}
 => {{1,4,7},{2,5},{3,6,8}}
 => {{1,8},{2,6,7},{3,4,5}}
 => ? = 9
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,4,5,6,7,8},{3}}
 => {{1,3,5,7,8},{2,4,6}}
 => {{1,7,8},{2,3,4,5,6}}
 => ? = 5
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,5,6,7,8},{4}}
 => {{1,3,5},{2,4,6,7,8}}
 => {{1,6,7,8},{2,3,4,5}}
 => ? = 4
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,6,7,8},{5}}
 => {{1,3,5,6,7,8},{2,4}}
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 3

Description

The lcb 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 '''lcb''' (left-closer-bigger) of

$S$ is given by a pair

$i < j$ such that

$j = \operatorname{max} B_b$ and

$i \in B_a$ for

$a > b$ .

Matching statistic: St000609

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
[1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 0
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 2
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,3},{2,4}}
 => {{1},{2,3,4}}
 => 2
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 1
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,4},{2},{3},{5}}
 => {{1},{2},{3,4},{5}}
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => {{1},{2,5},{3},{4}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => {{1},{2},{3,5},{4}}
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => {{1},{2},{3},{4,5}}
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,4},{2,5},{3}}
 => {{1},{2},{3,4,5}}
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1},{2,4},{3,5}}
 => {{1},{2,4,5},{3}}
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1,4},{2},{3,5}}
 => {{1},{2,5},{3,4}}
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => 3
[1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2},{3,4,5,6,7,8}}
 => {{1,2,3,4,5,6},{7},{8}}
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 0
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,5,6,7,8},{4}}
 => {{1,3,5},{2,4,6,7},{8}}
 => ?
 => ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,6,7,8},{5}}
 => {{1,3,5,6,7},{2,4},{8}}
 => {{1,6,7},{2,3,4,5},{8}}
 => ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,7,8},{6}}
 => {{1,3},{2,4,5,6,7},{8}}
 => {{1,5,6,7},{2,3,4},{8}}
 => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => {{1},{2,3,4,5,6,7},{8}}
 => {{1},{2,3,4,5,6,7},{8}}
 => {{1,3,4,5,6,7},{2},{8}}
 => ? = 0
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6,7},{2},{8}}
 => {{1,4,5,6,7},{2,3},{8}}
 => ? = 1
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,3,5,6,7,8},{2},{4}}
 => {{1,4,7},{2,5},{3,6,8}}
 => {{1},{2,8},{3,4,5,6,7}}
 => ? = 9
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,5,6,7,8},{3,4}}
 => {{1,3,5,6},{2,4,7,8}}
 => {{1,6,8},{2,3,4,5,7}}
 => ? = 4
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,5,6,7,8},{4}}
 => {{1,3,5},{2,4,6,7,8}}
 => {{1,7,8},{2,3,4,5,6}}
 => ? = 4
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,6,7,8},{4,5}}
 => {{1,3,6,7,8},{2,4,5}}
 => {{1,5,7,8},{2,3,4,6}}
 => ? = 3
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,6,7,8},{5}}
 => {{1,3,5,6,7,8},{2,4}}
 => {{1,6,7,8},{2,3,4,5}}
 => ? = 3
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,7,8},{5,6}}
 => {{1,3,4},{2,5,6,7,8}}
 => {{1,4,6,7,8},{2,3,5}}
 => ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,7,8},{6}}
 => {{1,3},{2,4,5,6,7,8}}
 => {{1,5,6,7,8},{2,3,4}}
 => ? = 2

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

Matching statistic: St001843

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001843: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%

Values

[1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => 0
[1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => 0
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,2},{3}}
 => 0
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => 0
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 0
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 0
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 2
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 2
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 0
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => 1
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,3},{2},{4},{5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,4},{2},{3},{5}}
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1,2,4},{3},{5}}
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2,3},{5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3,4},{2},{5}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,2,5},{3},{4}}
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,5},{2,3},{4}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => 3
[1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2},{3,4,5,6,7,8}}
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 0
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,4,5,6,7,8},{3}}
 => {{1,2,3,4,5,7},{6},{8}}
 => ? = 5
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,5,6,7,8},{4}}
 => {{1,2,3,4,6,7},{5},{8}}
 => ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,6,7,8},{5}}
 => {{1,2,3,5,6,7},{4},{8}}
 => ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,7,8},{6}}
 => {{1,2,4,5,6,7},{3},{8}}
 => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => {{1},{2,3,4,5,6,7},{8}}
 => {{1},{2,3,4,5,6,7},{8}}
 => ? = 0
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6,7},{2},{8}}
 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,7,8}}
 => {{1,2,3,4,5,6,7},{8}}
 => ? = 0
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2},{3,4,5,6,7,8}}
 => {{1,2,3,4,5,6},{7,8}}
 => ? = 0
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,4,5,6,7,8},{2},{3}}
 => {{1,2,3,4,5,8},{6},{7}}
 => ? = 10
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,3,5,6,7,8},{2},{4}}
 => {{1,2,3,4,6,8},{5},{7}}
 => ? = 9
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,3,4,5,6,7,8},{2}}
 => {{1,2,3,4,5,6,8},{7}}
 => ? = 6
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,4,5,6,7,8},{2,3}}
 => {{1,2,3,4,5,8},{6,7}}
 => ? = 5
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,4,5,6,7,8},{3}}
 => {{1,2,3,4,5,7,8},{6}}
 => ? = 5
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,5,6,7,8},{3,4}}
 => {{1,2,3,4,7,8},{5,6}}
 => ? = 4
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,5,6,7,8},{4}}
 => {{1,2,3,4,6,7,8},{5}}
 => ? = 4
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,6,7,8},{4,5}}
 => ? = 3
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,5,6,7,8},{4}}
 => ? = 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => {{1,2,3,4,5,6},{7,8}}
 => {{1,2},{3,4,5,6,7,8}}
 => ? = 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,7,8},{5,6}}
 => {{1,2,5,6,7,8},{3,4}}
 => ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,7,8},{6}}
 => {{1,2,4,5,6,7,8},{3}}
 => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => {{1,2,3,4,5,6,7},{8}}
 => {{1},{2,3,4,5,6,7,8}}
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => {{1,8},{2,3,4,5,6,7}}
 => {{1,8},{2,3,4,5,6,7}}
 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,8},{6,7}}
 => {{1,4,5,6,7,8},{2,3}}
 => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,8},{7}}
 => {{1,3,4,5,6,7,8},{2}}
 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7,8}}
 => {{1,2,3,4,5,6,7,8}}
 => ? = 0

Description

The Z-index of a set partition. The Mahonian representation of a set partition

$\{B_1,\dots,B_k\}$ of

$\{1,\dots,n\}$ is the restricted growth word

$w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting

$w_i$ to the index of the block containing

$i$ . The Z-index of

$w$ equals

$\sum_{i < j} w_{i,j},$ where

$w_{i,j}$ is the word obtained from

$w$ by removing all letters different from

$i$ and

$j$ .

Matching statistic: St000359
(load all 2 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%

Values

[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,1,3,2] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,2,1,3] => [3,2,4,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,4,2,3] => [1,4,5,3,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,2,4,3] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,3,2,4] => [1,4,3,5,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,4,1,2,3] => [3,4,5,2,1] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,1,2,4,3] => [2,3,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,3,1,2,4] => [3,4,2,5,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,3,1,2] => [4,5,3,2,1] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [1,2,7,6,3,4,5] => [1,2,5,6,7,4,3] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [1,2,6,5,3,4,7] => [1,2,5,6,4,3,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [1,2,7,3,4,6,5] => [1,2,4,5,7,6,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,5,3] => [1,2,7,5,3,4,6] => [1,2,5,6,4,7,3] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [1,2,7,6,5,3,4] => [1,2,6,7,5,4,3] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [1,2,6,3,5,4,7] => [1,2,4,6,5,3,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [1,2,7,3,5,4,6] => [1,2,4,6,5,7,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [1,2,7,6,3,5,4] => [1,2,5,7,6,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,4,3,7] => [1,2,6,4,3,5,7] => [1,2,5,4,6,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,4,7,3] => [1,2,7,3,6,4,5] => [1,2,4,6,7,5,3] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,4,3] => [1,2,7,4,3,5,6] => [1,2,5,4,6,7,3] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,6,4,3] => [1,2,7,6,4,3,5] => [1,2,6,5,7,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [1,2,7,3,6,5,4] => [1,2,4,7,6,5,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,5,7,4,3] => [1,2,7,4,3,6,5] => [1,2,5,4,7,6,3] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,5,4,3] => [1,2,7,5,4,3,6] => [1,2,6,5,4,7,3] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => [1,3,2,5,6,7,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,2,5,7,6,4] => [1,3,2,7,6,4,5] => [1,3,2,6,7,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,3,2,6,5,7,4] => [1,3,2,7,4,6,5] => [1,3,2,5,7,6,4] => ? = 1

Description

The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$23\!\!-\!\!1$ .

Matching statistic: St000355
(load all 2 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 64%

Values

[1,0,1,0]
 => [1,2] => [2,1] => 0
[1,1,0,0]
 => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,3,4] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,1,2,4] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,3,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,2,4,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,4,2,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,2,3,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,3,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,3,2,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,2,1,4,5] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [5,2,1,3,4] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,1,3,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,3,4,5] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [5,4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [7,5,4,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [6,5,4,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [5,4,6,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [5,6,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,5,4] => [6,4,5,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [6,5,7,3,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [6,7,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [7,5,4,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [6,5,4,3,7,2,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [5,4,3,6,7,2,1] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [7,4,3,5,6,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [5,6,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,5,3] => [6,4,3,5,7,2,1] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [4,3,5,6,7,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [7,4,5,3,6,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [6,4,5,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [4,5,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,4,3,7] => [7,5,3,4,6,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,4,7,3] => [5,4,6,3,7,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,4,3] => [6,5,3,4,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,6,4,3] => [5,3,4,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [4,5,6,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,5,7,4,3] => [5,6,3,4,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,5,4,3] => [6,3,4,5,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [6,5,7,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,6,4,7] => [7,5,4,6,2,3,1] => ? = 1

Description

The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$21\!\!-\!\!3$ .

Matching statistic: St001330
(load all 6 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 21%●distinct values known / distinct values provided: 18%

Values

[1,0,1,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 0 + 2
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 1 + 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2 + 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 1 + 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 4 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(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)
 => ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 2 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
 => ? = 3 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 2 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(0,6),(1,2),(1,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
 => ? = 3 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 3 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St000441

Mapping

Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 18%

Values

[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 0
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 0
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 0
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 0
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? = 1
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 0
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 0
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [6,5,3,4,2,1,8,7] => ? = 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 0
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => [2,1,10,7,5,9,4,8,6,3] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,5,8,7,6,4,3] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,6,7,8,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => [2,1,10,7,9,6,4,8,5,3] => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,6,7,5,4,3] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [3,4,2,1,7,9,6,10,8,5] => [4,3,2,1,10,9,7,8,6,5] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [3,4,2,1,8,9,10,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [3,5,2,6,4,1,9,10,8,7] => [6,5,3,4,2,1,10,9,8,7] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [3,5,2,7,4,8,6,1,10,9] => [8,5,3,7,2,6,4,1,10,9] => ? = 2
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 4
[1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [3,6,2,7,8,5,4,1,10,9] => [8,7,3,6,5,4,2,1,10,9] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 3
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 3
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [4,5,6,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [4,5,6,3,2,1,9,10,8,7] => [6,5,4,3,2,1,10,9,8,7] => 0
[1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [4,5,7,3,2,8,6,1,10,9] => [8,5,7,4,2,6,3,1,10,9] => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [4,6,7,3,8,5,2,1,10,9] => [8,7,6,4,5,3,2,1,10,9] => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [5,6,7,8,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 1
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,4,3,6,5,12,11,9,10,8,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
 => [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,4,3,10,9,7,8,6,5,12,11] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,4,3,12,9,7,11,6,10,8,5] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
 => [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,4,3,12,11,7,10,9,8,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
 => [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,4,3,12,9,11,8,6,10,7,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,4,3,12,11,10,8,9,7,6,5] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
 => [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
 => [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,6,5,4,3,12,11,9,10,8,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
 => [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
 => [2,1,6,7,8,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
 => [2,1,6,7,8,5,4,3,11,12,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
 => [2,1,6,7,9,5,4,10,8,3,12,11] => [2,1,10,7,9,6,4,8,5,3,12,11] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => [2,1,12,7,9,6,4,11,5,10,8,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [2,1,7,8,9,10,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,8,9,10,11,12,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 0
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
 => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 0

Description

The number of successions of a permutation. A succession of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)+1 = \pi(i+1)$ . Successions are also known as ''small ascents'' or ''1-rises''.

The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000058The order of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000516The number of stretching pairs of a permutation. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.