- FindStat

Your data matches 111 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000459
(load all 5 compositions to match this statistic)

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

Matching statistic: St000870
(load all 5 compositions to match this statistic)

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells

$(i,i)$ of a partition.

Matching statistic: St000806
(load all 9 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => [1,1] => 3 = 1 + 2
[1,1] => 11 => [1,1,1] => 4 = 2 + 2
[2] => 10 => [1,2] => 4 = 2 + 2
[1,1,1] => 111 => [1,1,1,1] => 5 = 3 + 2
[1,2] => 110 => [1,1,2] => 5 = 3 + 2
[2,1] => 101 => [1,2,1] => 5 = 3 + 2
[3] => 100 => [1,3] => 5 = 3 + 2
[1,1,1,1] => 1111 => [1,1,1,1,1] => 6 = 4 + 2
[1,1,2] => 1110 => [1,1,1,2] => 6 = 4 + 2
[1,2,1] => 1101 => [1,1,2,1] => 6 = 4 + 2
[1,3] => 1100 => [1,1,3] => 6 = 4 + 2
[2,1,1] => 1011 => [1,2,1,1] => 6 = 4 + 2
[2,2] => 1010 => [1,2,2] => 5 = 3 + 2
[3,1] => 1001 => [1,3,1] => 6 = 4 + 2
[4] => 1000 => [1,4] => 6 = 4 + 2
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => 7 = 5 + 2
[1,1,1,2] => 11110 => [1,1,1,1,2] => 7 = 5 + 2
[1,1,2,1] => 11101 => [1,1,1,2,1] => 7 = 5 + 2
[1,1,3] => 11100 => [1,1,1,3] => 7 = 5 + 2
[1,2,1,1] => 11011 => [1,1,2,1,1] => 7 = 5 + 2
[1,2,2] => 11010 => [1,1,2,2] => 6 = 4 + 2
[1,3,1] => 11001 => [1,1,3,1] => 7 = 5 + 2
[1,4] => 11000 => [1,1,4] => 7 = 5 + 2
[2,1,1,1] => 10111 => [1,2,1,1,1] => 7 = 5 + 2
[2,1,2] => 10110 => [1,2,1,2] => 7 = 5 + 2
[2,2,1] => 10101 => [1,2,2,1] => 6 = 4 + 2
[2,3] => 10100 => [1,2,3] => 6 = 4 + 2
[3,1,1] => 10011 => [1,3,1,1] => 7 = 5 + 2
[3,2] => 10010 => [1,3,2] => 6 = 4 + 2
[4,1] => 10001 => [1,4,1] => 7 = 5 + 2
[5] => 10000 => [1,5] => 7 = 5 + 2

Description

The semiperimeter of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.

Matching statistic: St000216
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The absolute length of a permutation. The absolute length of a permutation

$\sigma$ of length

$n$ is the shortest

$k$ such that

$\sigma = t_1 \dots t_k$ for transpositions

$t_i$ . Also, this is equal to

$n$ minus the number of cycles of

$\sigma$ .

Matching statistic: St000876

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => 1 => 1 => 1
[1,1] => 11 => 11 => 11 => 2
[2] => 10 => 01 => 01 => 2
[1,1,1] => 111 => 111 => 111 => 3
[1,2] => 110 => 011 => 011 => 3
[2,1] => 101 => 110 => 011 => 3
[3] => 100 => 010 => 001 => 3
[1,1,1,1] => 1111 => 1111 => 1111 => 4
[1,1,2] => 1110 => 0111 => 0111 => 4
[1,2,1] => 1101 => 1110 => 0111 => 4
[1,3] => 1100 => 0110 => 0011 => 4
[2,1,1] => 1011 => 1101 => 0111 => 4
[2,2] => 1010 => 0101 => 0101 => 3
[3,1] => 1001 => 1100 => 0011 => 4
[4] => 1000 => 0100 => 0001 => 4
[1,1,1,1,1] => 11111 => 11111 => 11111 => 5
[1,1,1,2] => 11110 => 01111 => 01111 => 5
[1,1,2,1] => 11101 => 11110 => 01111 => 5
[1,1,3] => 11100 => 01110 => 00111 => 5
[1,2,1,1] => 11011 => 11101 => 01111 => 5
[1,2,2] => 11010 => 01101 => 01011 => 4
[1,3,1] => 11001 => 11100 => 00111 => 5
[1,4] => 11000 => 01100 => 00011 => 5
[2,1,1,1] => 10111 => 11011 => 01111 => 5
[2,1,2] => 10110 => 01011 => 01011 => 4
[2,2,1] => 10101 => 11010 => 00111 => 5
[2,3] => 10100 => 01010 => 00101 => 4
[3,1,1] => 10011 => 11001 => 00111 => 5
[3,2] => 10010 => 01001 => 00101 => 4
[4,1] => 10001 => 11000 => 00011 => 5
[5] => 10000 => 01000 => 00001 => 5

Description

The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as

$(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$ , where

$\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is,

$\ell + m$ if the middle Dyck word is empty and

$\ell + 1 + m$ otherwise.

Matching statistic: St001004

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a

$321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

Matching statistic: St000672
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Matching statistic: St001012
(load all 5 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001023

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001023: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001179
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001179: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.

The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000209Maximum difference of elements in cycles. St001424The number of distinct squares in a binary word. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000080The rank of the poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000004The major index of a permutation. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000653The last descent of a permutation. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents 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

$K[x]/(x^n)$ . St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001894The depth of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000325The width of the tree associated to a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000470The number of runs in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St001926Sparre Andersen's position of the maximum of a signed permutation. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000782The indicator function of whether a given perfect matching is an L & P matching. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St001556The number of inversions of the third entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.