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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001499
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Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001499: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001499: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001167
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St001167: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St001167: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 1 - 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001199
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001199: Dyck paths ā¶ ā¤Result quality: 93% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001199: Dyck paths ā¶ ā¤Result quality: 93% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[8,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[5,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[9,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[8,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[7,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[6,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[10,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[9,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[8,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[7,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,6]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[11,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[10,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[9,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[8,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,6]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[12,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[11,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[10,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[9,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[8,6]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[7,7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[13,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[12,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[11,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[10,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[9,6]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[8,7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[14,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[13,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[12,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[11,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[10,6]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[9,7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[15,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
[14,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000371
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000371: Permutations ā¶ ā¤Result quality: 89% āvalues known / values provided: 89%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000371: Permutations ā¶ ā¤Result quality: 89% āvalues known / values provided: 89%ādistinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3 - 1
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 3 - 1
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3 - 1
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 3 - 1
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 3 - 1
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 3 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 3 - 1
[5,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3 - 1
[4,3,3,2,1,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 3 - 1
[4,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 3 - 1
[4,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 3 - 1
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 4 - 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 3 - 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 3 - 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 3 - 1
[6,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3 - 1
[5,3,3,2,1,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 3 - 1
[5,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 3 - 1
[5,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 3 - 1
[4,4,4,2,1]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 3 - 1
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 4 - 1
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 3 - 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 3 - 1
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => ? = 4 - 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 3 - 1
[3,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 3 - 1
[7,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3 - 1
[6,3,3,2,1,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 3 - 1
[6,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 3 - 1
[6,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 3 - 1
[5,4,4,2,1]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 3 - 1
[5,3,3,3,1,1]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 4 - 1
[5,3,3,2,2,1]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 3 - 1
[5,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 3 - 1
[4,4,4,3,1]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => ? = 4 - 1
[4,4,4,2,2]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => ? = 3 - 1
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => ? = 4 - 1
[4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 3 - 1
[4,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 3 - 1
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 5 - 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 4 - 1
[8,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3 - 1
[7,3,3,2,1,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 3 - 1
[7,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 3 - 1
[7,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 3 - 1
[6,4,4,2,1]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 3 - 1
[6,3,3,3,1,1]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 4 - 1
[6,3,3,2,2,1]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 3 - 1
[6,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 3 - 1
[5,5,5,2]
=> [5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => ? = 3 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000372
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000372: Permutations ā¶ ā¤Result quality: 67% āvalues known / values provided: 67%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000372: Permutations ā¶ ā¤Result quality: 67% āvalues known / values provided: 67%ādistinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0 = 1 - 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1 = 2 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => 0 = 1 - 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1 = 2 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 2 = 3 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 1 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 2 - 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 1 - 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 1 - 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 1 - 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => ? = 2 - 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 2 - 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 3 - 1
[5,5,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[5,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 1 - 1
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 1 - 1
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 1 - 1
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[4,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => ? = 2 - 1
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => ? = 2 - 1
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 2 - 1
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 2 - 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,7,1] => ? = 3 - 1
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 3 - 1
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 3 - 1
[6,6,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1 - 1
[6,5,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[6,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 1 - 1
[6,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 1 - 1
[6,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 1 - 1
[6,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[5,5,2,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 2 - 1
[5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => ? = 2 - 1
[5,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => ? = 2 - 1
[5,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 2 - 1
[4,4,3,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,7,1] => ? = 2 - 1
[4,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 3 - 1
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 2 - 1
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,7,1] => ? = 3 - 1
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 3 - 1
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,1,4,7] => ? = 3 - 1
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => ? = 3 - 1
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 3 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 3 - 1
[7,7]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[7,6,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1 - 1
[7,5,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[7,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 1 - 1
[7,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 1 - 1
Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Matching statistic: St000779
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000779: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000779: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1 = 2 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 2 - 1
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1 = 2 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 3 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[3,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2 - 1
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 2 - 1
[4,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[4,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2 - 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 - 1
[4,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2 - 1
[4,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3 - 1
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 - 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 3 - 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? = 2 - 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 3 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 2 - 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 2 - 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => ? = 3 - 1
[5,5,2]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2 - 1
[5,5,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 - 1
[5,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[5,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2 - 1
[5,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 - 1
[5,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2 - 1
[5,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3 - 1
[5,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 - 1
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[5,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[5,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 2 - 1
Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St000872
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000872: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000872: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1 = 2 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1 = 2 - 1
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1 = 2 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 3 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 3 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 2 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[3,3,3,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 2 - 1
[3,3,2,2]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 3 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 3 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 2 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ? = 3 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ? = 2 - 1
[4,4,3]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 2 - 1
[4,4,2,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 2 - 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 - 1
[4,3,3,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 2 - 1
[4,3,2,2]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 3 - 1
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 - 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 - 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 3 - 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 2 - 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => ? = 3 - 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => ? = 2 - 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ? = 3 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => ? = 2 - 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ? = 3 - 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ? = 2 - 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => ? = 3 - 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => ? = 3 - 1
[5,5,2]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ? = 2 - 1
[5,5,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[5,4,3]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 2 - 1
[5,4,2,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 2 - 1
[5,4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 - 1
[5,3,3,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 2 - 1
[5,3,2,2]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 3 - 1
[5,3,2,1,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 - 1
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 - 1
[5,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 3 - 1
[5,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 2 - 1
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 2 - 1
Description
The number of very big descents of a permutation.
A very big descent of a permutation $\pi$ is an index $i$ such that $\pi_i - \pi_{i+1} > 2$.
For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General $r$-descents were for example be studied in [1, Section 2].
Matching statistic: St000099
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000099: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000099: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[3,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 2
[4,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 2
[4,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[4,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2
[4,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 3
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? = 2
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 3
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 2
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 2
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ? = 3
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => ? = 3
[5,5,2]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2
[5,5,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[5,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 2
[5,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2
[5,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[5,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2
[5,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3
[5,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[5,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3
[5,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 2
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000023
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000023: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000023: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1 = 2 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0 = 1 - 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 2 - 1
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0 = 1 - 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1 = 2 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 3 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1 = 2 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[3,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2 - 1
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 - 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 2 - 1
[4,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[4,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2 - 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 - 1
[4,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2 - 1
[4,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3 - 1
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 - 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 3 - 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? = 2 - 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 3 - 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 2 - 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 - 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 2 - 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ? = 3 - 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => ? = 3 - 1
[5,5,2]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2 - 1
[5,5,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 - 1
[5,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 2 - 1
[5,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 2 - 1
[5,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 - 1
[5,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 2 - 1
[5,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 3 - 1
[5,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 - 1
[5,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1 - 1
[5,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 3 - 1
[5,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 2 - 1
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1 - 1
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1 - 1
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 2 - 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000442
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00142: Dyck paths āpromotionā¶ Dyck paths
Mp00228: Dyck paths āreflect parallelogram polyominoā¶ Dyck paths
St000442: Dyck paths ā¶ ā¤Result quality: 28% āvalues known / values provided: 28%ādistinct values known / distinct values provided: 83%
Mp00142: Dyck paths āpromotionā¶ Dyck paths
Mp00228: Dyck paths āreflect parallelogram polyominoā¶ Dyck paths
St000442: Dyck paths ā¶ ā¤Result quality: 28% āvalues known / values provided: 28%ādistinct values known / distinct values provided: 83%
Values
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ?
=> ? = 1
[7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 2
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 1
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 1
[6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ?
=> ? = 1
[6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 1
[6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 1
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ?
=> ? = 2
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 1
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 1
[4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 2
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 2
[2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ?
=> ? = 1
[9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ?
=> ?
=> ? = 1
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 1
[8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
Description
The maximal area to the right of an up step of a Dyck path.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000820The number of compositions obtained by rotating the composition. St000017The number of inversions of a standard tableau. St000782The indicator function of whether a given perfect matching is an L & P matching. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000251The number of nonsingleton blocks of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001969The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. St000648The number of 2-excedences of a permutation. St000842The breadth of a permutation. St001537The number of cyclic crossings of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000405The number of occurrences of the pattern 1324 in a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000516The number of stretching pairs of a permutation. St001513The number of nested exceedences of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St000624The normalized sum of the minimal distances to a greater element.
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