- FindStat

Your data matches 209 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001124
(load all 6 compositions to match this statistic)

Mapping

St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 0
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 1
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 1
[4,2]
 => 1
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 2
[3,1,1,1]
 => 1
[2,2,2]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 1
[4,3]
 => 1
[4,2,1]
 => 2
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 0
[7,1]
 => 1
[6,2]
 => 1
[6,1,1]
 => 1
[5,3]
 => 1
[5,2,1]
 => 2
[5,1,1,1]
 => 1

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000318
(load all 6 compositions to match this statistic)

Mapping

St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 2 = 0 + 2
[1,1]
 => 2 = 0 + 2
[3]
 => 2 = 0 + 2
[2,1]
 => 3 = 1 + 2
[1,1,1]
 => 2 = 0 + 2
[4]
 => 2 = 0 + 2
[3,1]
 => 3 = 1 + 2
[2,2]
 => 2 = 0 + 2
[2,1,1]
 => 3 = 1 + 2
[1,1,1,1]
 => 2 = 0 + 2
[5]
 => 2 = 0 + 2
[4,1]
 => 3 = 1 + 2
[3,2]
 => 3 = 1 + 2
[3,1,1]
 => 3 = 1 + 2
[2,2,1]
 => 3 = 1 + 2
[2,1,1,1]
 => 3 = 1 + 2
[1,1,1,1,1]
 => 2 = 0 + 2
[6]
 => 2 = 0 + 2
[5,1]
 => 3 = 1 + 2
[4,2]
 => 3 = 1 + 2
[4,1,1]
 => 3 = 1 + 2
[3,3]
 => 2 = 0 + 2
[3,2,1]
 => 4 = 2 + 2
[3,1,1,1]
 => 3 = 1 + 2
[2,2,2]
 => 2 = 0 + 2
[2,2,1,1]
 => 3 = 1 + 2
[2,1,1,1,1]
 => 3 = 1 + 2
[1,1,1,1,1,1]
 => 2 = 0 + 2
[7]
 => 2 = 0 + 2
[6,1]
 => 3 = 1 + 2
[5,2]
 => 3 = 1 + 2
[5,1,1]
 => 3 = 1 + 2
[4,3]
 => 3 = 1 + 2
[4,2,1]
 => 4 = 2 + 2
[4,1,1,1]
 => 3 = 1 + 2
[3,3,1]
 => 3 = 1 + 2
[3,2,2]
 => 3 = 1 + 2
[3,2,1,1]
 => 4 = 2 + 2
[3,1,1,1,1]
 => 3 = 1 + 2
[2,2,2,1]
 => 3 = 1 + 2
[2,2,1,1,1]
 => 3 = 1 + 2
[2,1,1,1,1,1]
 => 3 = 1 + 2
[1,1,1,1,1,1,1]
 => 2 = 0 + 2
[8]
 => 2 = 0 + 2
[7,1]
 => 3 = 1 + 2
[6,2]
 => 3 = 1 + 2
[6,1,1]
 => 3 = 1 + 2
[5,3]
 => 3 = 1 + 2
[5,2,1]
 => 4 = 2 + 2
[5,1,1,1]
 => 3 = 1 + 2

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000292
(load all 32 compositions to match this statistic)

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 100 => 0
[1,1]
 => 110 => 0
[3]
 => 1000 => 0
[2,1]
 => 1010 => 1
[1,1,1]
 => 1110 => 0
[4]
 => 10000 => 0
[3,1]
 => 10010 => 1
[2,2]
 => 1100 => 0
[2,1,1]
 => 10110 => 1
[1,1,1,1]
 => 11110 => 0
[5]
 => 100000 => 0
[4,1]
 => 100010 => 1
[3,2]
 => 10100 => 1
[3,1,1]
 => 100110 => 1
[2,2,1]
 => 11010 => 1
[2,1,1,1]
 => 101110 => 1
[1,1,1,1,1]
 => 111110 => 0
[6]
 => 1000000 => 0
[5,1]
 => 1000010 => 1
[4,2]
 => 100100 => 1
[4,1,1]
 => 1000110 => 1
[3,3]
 => 11000 => 0
[3,2,1]
 => 101010 => 2
[3,1,1,1]
 => 1001110 => 1
[2,2,2]
 => 11100 => 0
[2,2,1,1]
 => 110110 => 1
[2,1,1,1,1]
 => 1011110 => 1
[1,1,1,1,1,1]
 => 1111110 => 0
[7]
 => 10000000 => 0
[6,1]
 => 10000010 => 1
[5,2]
 => 1000100 => 1
[5,1,1]
 => 10000110 => 1
[4,3]
 => 101000 => 1
[4,2,1]
 => 1001010 => 2
[4,1,1,1]
 => 10001110 => 1
[3,3,1]
 => 110010 => 1
[3,2,2]
 => 101100 => 1
[3,2,1,1]
 => 1010110 => 2
[3,1,1,1,1]
 => 10011110 => 1
[2,2,2,1]
 => 111010 => 1
[2,2,1,1,1]
 => 1101110 => 1
[2,1,1,1,1,1]
 => 10111110 => 1
[1,1,1,1,1,1,1]
 => 11111110 => 0
[8]
 => 100000000 => 0
[7,1]
 => 100000010 => 1
[6,2]
 => 10000100 => 1
[6,1,1]
 => 100000110 => 1
[5,3]
 => 1001000 => 1
[5,2,1]
 => 10001010 => 2
[5,1,1,1]
 => 100001110 => 1

Description

The number of ascents of a binary word.

Matching statistic: St000291
(load all 37 compositions to match this statistic)

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 100 => 1 = 0 + 1
[1,1]
 => 110 => 1 = 0 + 1
[3]
 => 1000 => 1 = 0 + 1
[2,1]
 => 1010 => 2 = 1 + 1
[1,1,1]
 => 1110 => 1 = 0 + 1
[4]
 => 10000 => 1 = 0 + 1
[3,1]
 => 10010 => 2 = 1 + 1
[2,2]
 => 1100 => 1 = 0 + 1
[2,1,1]
 => 10110 => 2 = 1 + 1
[1,1,1,1]
 => 11110 => 1 = 0 + 1
[5]
 => 100000 => 1 = 0 + 1
[4,1]
 => 100010 => 2 = 1 + 1
[3,2]
 => 10100 => 2 = 1 + 1
[3,1,1]
 => 100110 => 2 = 1 + 1
[2,2,1]
 => 11010 => 2 = 1 + 1
[2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,1,1,1,1]
 => 111110 => 1 = 0 + 1
[6]
 => 1000000 => 1 = 0 + 1
[5,1]
 => 1000010 => 2 = 1 + 1
[4,2]
 => 100100 => 2 = 1 + 1
[4,1,1]
 => 1000110 => 2 = 1 + 1
[3,3]
 => 11000 => 1 = 0 + 1
[3,2,1]
 => 101010 => 3 = 2 + 1
[3,1,1,1]
 => 1001110 => 2 = 1 + 1
[2,2,2]
 => 11100 => 1 = 0 + 1
[2,2,1,1]
 => 110110 => 2 = 1 + 1
[2,1,1,1,1]
 => 1011110 => 2 = 1 + 1
[1,1,1,1,1,1]
 => 1111110 => 1 = 0 + 1
[7]
 => 10000000 => 1 = 0 + 1
[6,1]
 => 10000010 => 2 = 1 + 1
[5,2]
 => 1000100 => 2 = 1 + 1
[5,1,1]
 => 10000110 => 2 = 1 + 1
[4,3]
 => 101000 => 2 = 1 + 1
[4,2,1]
 => 1001010 => 3 = 2 + 1
[4,1,1,1]
 => 10001110 => 2 = 1 + 1
[3,3,1]
 => 110010 => 2 = 1 + 1
[3,2,2]
 => 101100 => 2 = 1 + 1
[3,2,1,1]
 => 1010110 => 3 = 2 + 1
[3,1,1,1,1]
 => 10011110 => 2 = 1 + 1
[2,2,2,1]
 => 111010 => 2 = 1 + 1
[2,2,1,1,1]
 => 1101110 => 2 = 1 + 1
[2,1,1,1,1,1]
 => 10111110 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => 11111110 => 1 = 0 + 1
[8]
 => 100000000 => 1 = 0 + 1
[7,1]
 => 100000010 => 2 = 1 + 1
[6,2]
 => 10000100 => 2 = 1 + 1
[6,1,1]
 => 100000110 => 2 = 1 + 1
[5,3]
 => 1001000 => 2 = 1 + 1
[5,2,1]
 => 10001010 => 3 = 2 + 1
[5,1,1,1]
 => 100001110 => 2 = 1 + 1

Description

The number of descents of a binary word.

Matching statistic: St000390
(load all 34 compositions to match this statistic)

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 100 => 1 = 0 + 1
[1,1]
 => 110 => 1 = 0 + 1
[3]
 => 1000 => 1 = 0 + 1
[2,1]
 => 1010 => 2 = 1 + 1
[1,1,1]
 => 1110 => 1 = 0 + 1
[4]
 => 10000 => 1 = 0 + 1
[3,1]
 => 10010 => 2 = 1 + 1
[2,2]
 => 1100 => 1 = 0 + 1
[2,1,1]
 => 10110 => 2 = 1 + 1
[1,1,1,1]
 => 11110 => 1 = 0 + 1
[5]
 => 100000 => 1 = 0 + 1
[4,1]
 => 100010 => 2 = 1 + 1
[3,2]
 => 10100 => 2 = 1 + 1
[3,1,1]
 => 100110 => 2 = 1 + 1
[2,2,1]
 => 11010 => 2 = 1 + 1
[2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,1,1,1,1]
 => 111110 => 1 = 0 + 1
[6]
 => 1000000 => 1 = 0 + 1
[5,1]
 => 1000010 => 2 = 1 + 1
[4,2]
 => 100100 => 2 = 1 + 1
[4,1,1]
 => 1000110 => 2 = 1 + 1
[3,3]
 => 11000 => 1 = 0 + 1
[3,2,1]
 => 101010 => 3 = 2 + 1
[3,1,1,1]
 => 1001110 => 2 = 1 + 1
[2,2,2]
 => 11100 => 1 = 0 + 1
[2,2,1,1]
 => 110110 => 2 = 1 + 1
[2,1,1,1,1]
 => 1011110 => 2 = 1 + 1
[1,1,1,1,1,1]
 => 1111110 => 1 = 0 + 1
[7]
 => 10000000 => 1 = 0 + 1
[6,1]
 => 10000010 => 2 = 1 + 1
[5,2]
 => 1000100 => 2 = 1 + 1
[5,1,1]
 => 10000110 => 2 = 1 + 1
[4,3]
 => 101000 => 2 = 1 + 1
[4,2,1]
 => 1001010 => 3 = 2 + 1
[4,1,1,1]
 => 10001110 => 2 = 1 + 1
[3,3,1]
 => 110010 => 2 = 1 + 1
[3,2,2]
 => 101100 => 2 = 1 + 1
[3,2,1,1]
 => 1010110 => 3 = 2 + 1
[3,1,1,1,1]
 => 10011110 => 2 = 1 + 1
[2,2,2,1]
 => 111010 => 2 = 1 + 1
[2,2,1,1,1]
 => 1101110 => 2 = 1 + 1
[2,1,1,1,1,1]
 => 10111110 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => 11111110 => 1 = 0 + 1
[8]
 => 100000000 => 1 = 0 + 1
[7,1]
 => 100000010 => 2 = 1 + 1
[6,2]
 => 10000100 => 2 = 1 + 1
[6,1,1]
 => 100000110 => 2 = 1 + 1
[5,3]
 => 1001000 => 2 = 1 + 1
[5,2,1]
 => 10001010 => 3 = 2 + 1
[5,1,1,1]
 => 100001110 => 2 = 1 + 1

Description

The number of runs of ones in a binary word.

Matching statistic: St000288
(load all 5 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 10 => 1 = 0 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 10 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 100 => 1 = 0 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 11 => 2 = 1 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 100 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1000 => 1 = 0 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 110 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 100 => 1 = 0 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 110 => 2 = 1 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1000 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 10000 => 1 = 0 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1100 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 101 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 110 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 110 => 2 = 1 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1100 => 2 = 1 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 10000 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => 100000 => 1 = 0 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 11000 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 1010 => 2 = 1 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1100 => 2 = 1 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1000 => 1 = 0 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 111 => 3 = 2 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1100 => 2 = 1 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1000 => 1 = 0 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 1100 => 2 = 1 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 11000 => 2 = 1 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 100000 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => 1000000 => 1 = 0 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => 110000 => 2 = 1 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 10100 => 2 = 1 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 11000 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1001 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1110 => 3 = 2 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1100 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1100 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 1010 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1110 => 3 = 2 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 11000 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1100 => 2 = 1 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 11000 => 2 = 1 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => 110000 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 1000000 => 1 = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => 10000000 => 1 = 0 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => 1100000 => 2 = 1 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => 101000 => 2 = 1 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => 110000 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 10010 => 2 = 1 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 11100 => 3 = 2 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 11000 => 2 = 1 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St001918

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the cyclic sieving polynomial corresponding to an integer partition. Let

$\lambda$ be an integer partition of

$n$ and let

$N$ be the least common multiple of the parts of

$\lambda$ . Fix an arbitrary permutation

$\pi$ of cycle type

$\lambda$ . Then

$\pi$ induces a cyclic action of order

$N$ on

$\{1,\dots,n\}$ . The corresponding character can be identified with the cyclic sieving polynomial

$C_\lambda(q)$ of this action, modulo

$q^N-1$ . Explicitly, it is

$\sum_{p\in\lambda} [p]_{q^{N/p}},$ where

$[p]_q = 1+\dots+q^{p-1}$ is the

$q$ -integer. This statistic records the degree of

$C_\lambda(q)$ . Equivalently, it equals

$\left(1 - \frac{1}{\lambda_1}\right) N,$ where

$\lambda_1$ is the largest part of

$\lambda$ . The statistic is undefined for the empty partition.

Matching statistic: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000147
(load all 2 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000668

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The least common multiple of the parts of the partition.

The following 199 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000708The product of the parts of an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000011The number of touch points (or returns) of a Dyck path. St000507The number of ascents of a standard tableau. St001280The number of parts of an integer partition that are at least two. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001581The achromatic number of a graph. St000159The number of distinct parts of the integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000306The bounce count of a Dyck path. St000676The number of odd rises of a Dyck path. St000662The staircase size of the code of a permutation. St000925The number of topologically connected components of a set partition. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St001029The size of the core of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000141The maximum drop size of a permutation. St000528The height of a poset. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000093The cardinality of a maximal independent set of vertices of a graph. St000527The width of the poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000312The number of leaves in a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000007The number of saliances of the permutation. St000702The number of weak deficiencies of a permutation. St000632The jump number of the poset. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000153The number of adjacent cycles of a permutation. St000069The number of maximal elements of a poset. St000619The number of cyclic descents of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000386The number of factors DDU in a Dyck path. St001732The number of peaks visible from the left. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000308The height of the tree associated to a permutation. St000809The reduced reflection length of the permutation. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000065The number of entries equal to -1 in an alternating sign matrix. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000237The number of small exceedances. St000389The number of runs of ones of odd length in a binary word. St000636The hull number of a graph. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001883The mutual visibility number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000201The number of leaf nodes in a binary tree. St001812The biclique partition number of a graph. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000647The number of big descents of a permutation. St000764The number of strong records in an integer composition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000761The number of ascents in an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000646The number of big ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000353The number of inner valleys of a permutation. St000862The number of parts of the shifted shape of a permutation. St000711The number of big exceedences of a permutation. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000388The number of orbits of vertices of a graph under automorphisms. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000317The cycle descent number of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001964The interval resolution global dimension of a poset. St001741The largest integer such that all patterns of this size are contained in the permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000779The tier of a permutation. St001470The cyclic holeyness of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000824The sum of the number of descents and the number of recoils of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001330The hat guessing number of a graph. St001863The number of weak excedances of a signed permutation. St001487The number of inner corners of a skew partition. St001864The number of excedances of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St000893The number of distinct diagonal sums of an alternating sign matrix. St001488The number of corners of a skew partition. St000451The length of the longest pattern of the form k 1 2.