- FindStat

Identifier

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

0 => 1 => [1] => [1] => 1

1 => 1 => [1] => [1] => 1

00 => 01 => [1,1] => [1,1] => 2

01 => 10 => [1,1] => [1,1] => 2

10 => 11 => [2] => [2] => 1

11 => 11 => [2] => [2] => 1

000 => 001 => [2,1] => [2,1] => 2

001 => 010 => [1,1,1] => [1,1,1] => 3

010 => 101 => [1,1,1] => [1,1,1] => 3

011 => 101 => [1,1,1] => [1,1,1] => 3

100 => 101 => [1,1,1] => [1,1,1] => 3

101 => 110 => [2,1] => [2,1] => 2

110 => 111 => [3] => [3] => 1

111 => 111 => [3] => [3] => 1

0000 => 0001 => [3,1] => [3,1] => 2

0001 => 0010 => [2,1,1] => [2,1,1] => 3

0010 => 0101 => [1,1,1,1] => [1,1,1,1] => 4

0011 => 0101 => [1,1,1,1] => [1,1,1,1] => 4

0100 => 1001 => [1,2,1] => [2,1,1] => 3

0101 => 1010 => [1,1,1,1] => [1,1,1,1] => 4

0110 => 1011 => [1,1,2] => [2,1,1] => 3

0111 => 1011 => [1,1,2] => [2,1,1] => 3

1000 => 1001 => [1,2,1] => [2,1,1] => 3

1001 => 1010 => [1,1,1,1] => [1,1,1,1] => 4

1010 => 1101 => [2,1,1] => [2,1,1] => 3

1011 => 1101 => [2,1,1] => [2,1,1] => 3

1100 => 1101 => [2,1,1] => [2,1,1] => 3

1101 => 1110 => [3,1] => [3,1] => 2

1110 => 1111 => [4] => [4] => 1

1111 => 1111 => [4] => [4] => 1

00000 => 00001 => [4,1] => [4,1] => 2

00001 => 00010 => [3,1,1] => [3,1,1] => 3

00010 => 00101 => [2,1,1,1] => [2,1,1,1] => 4

00011 => 00101 => [2,1,1,1] => [2,1,1,1] => 4

00100 => 01001 => [1,1,2,1] => [2,1,1,1] => 4

00101 => 01010 => [1,1,1,1,1] => [1,1,1,1,1] => 5

00110 => 01011 => [1,1,1,2] => [2,1,1,1] => 4

00111 => 01011 => [1,1,1,2] => [2,1,1,1] => 4

01000 => 10001 => [1,3,1] => [3,1,1] => 3

01001 => 10010 => [1,2,1,1] => [2,1,1,1] => 4

01010 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 5

01011 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 5

01100 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 5

01101 => 10110 => [1,1,2,1] => [2,1,1,1] => 4

01110 => 10111 => [1,1,3] => [3,1,1] => 3

01111 => 10111 => [1,1,3] => [3,1,1] => 3

10000 => 10001 => [1,3,1] => [3,1,1] => 3

10001 => 10010 => [1,2,1,1] => [2,1,1,1] => 4

10010 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 5

10011 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 5

10100 => 11001 => [2,2,1] => [2,2,1] => 3

10101 => 11010 => [2,1,1,1] => [2,1,1,1] => 4

10110 => 11011 => [2,1,2] => [2,2,1] => 3

10111 => 11011 => [2,1,2] => [2,2,1] => 3

11000 => 11001 => [2,2,1] => [2,2,1] => 3

11001 => 11010 => [2,1,1,1] => [2,1,1,1] => 4

11010 => 11101 => [3,1,1] => [3,1,1] => 3

11011 => 11101 => [3,1,1] => [3,1,1] => 3

11100 => 11101 => [3,1,1] => [3,1,1] => 3

11101 => 11110 => [4,1] => [4,1] => 2

11110 => 11111 => [5] => [5] => 1

11111 => 11111 => [5] => [5] => 1

000000 => 000001 => [5,1] => [5,1] => 2

000001 => 000010 => [4,1,1] => [4,1,1] => 3

000010 => 000101 => [3,1,1,1] => [3,1,1,1] => 4

000011 => 000101 => [3,1,1,1] => [3,1,1,1] => 4

000100 => 001001 => [2,1,2,1] => [2,2,1,1] => 4

000101 => 001010 => [2,1,1,1,1] => [2,1,1,1,1] => 5

000110 => 001011 => [2,1,1,2] => [2,2,1,1] => 4

000111 => 001011 => [2,1,1,2] => [2,2,1,1] => 4

001000 => 010001 => [1,1,3,1] => [3,1,1,1] => 4

001001 => 010010 => [1,1,2,1,1] => [2,1,1,1,1] => 5

001010 => 010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

001011 => 010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

001100 => 010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

001101 => 010110 => [1,1,1,2,1] => [2,1,1,1,1] => 5

001110 => 010111 => [1,1,1,3] => [3,1,1,1] => 4

001111 => 010111 => [1,1,1,3] => [3,1,1,1] => 4

010000 => 100001 => [1,4,1] => [4,1,1] => 3

010001 => 100010 => [1,3,1,1] => [3,1,1,1] => 4

010010 => 100101 => [1,2,1,1,1] => [2,1,1,1,1] => 5

010011 => 100101 => [1,2,1,1,1] => [2,1,1,1,1] => 5

010100 => 101001 => [1,1,1,2,1] => [2,1,1,1,1] => 5

010101 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

010110 => 101011 => [1,1,1,1,2] => [2,1,1,1,1] => 5

010111 => 101011 => [1,1,1,1,2] => [2,1,1,1,1] => 5

011000 => 101001 => [1,1,1,2,1] => [2,1,1,1,1] => 5

011001 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

011010 => 101101 => [1,1,2,1,1] => [2,1,1,1,1] => 5

011011 => 101101 => [1,1,2,1,1] => [2,1,1,1,1] => 5

011100 => 101101 => [1,1,2,1,1] => [2,1,1,1,1] => 5

011101 => 101110 => [1,1,3,1] => [3,1,1,1] => 4

011110 => 101111 => [1,1,4] => [4,1,1] => 3

011111 => 101111 => [1,1,4] => [4,1,1] => 3

100000 => 100001 => [1,4,1] => [4,1,1] => 3

100001 => 100010 => [1,3,1,1] => [3,1,1,1] => 4

100010 => 100101 => [1,2,1,1,1] => [2,1,1,1,1] => 5

100011 => 100101 => [1,2,1,1,1] => [2,1,1,1,1] => 5

100100 => 101001 => [1,1,1,2,1] => [2,1,1,1,1] => 5

100101 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

100110 => 101011 => [1,1,1,1,2] => [2,1,1,1,1] => 5

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$F_{1} = 2\ q$

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

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

$F_{4} = 2\ q + 2\ q^{2} + 8\ q^{3} + 4\ q^{4}$

$F_{5} = 2\ q + 2\ q^{2} + 12\ q^{3} + 10\ q^{4} + 6\ q^{5}$

$F_{6} = 2\ q + 2\ q^{2} + 16\ q^{3} + 16\ q^{4} + 22\ q^{5} + 6\ q^{6}$

$F_{7} = 2\ q + 2\ q^{2} + 20\ q^{3} + 22\ q^{4} + 48\ q^{5} + 24\ q^{6} + 10\ q^{7}$

$F_{8} = 2\ q + 2\ q^{2} + 24\ q^{3} + 28\ q^{4} + 84\ q^{5} + 56\ q^{6} + 50\ q^{7} + 10\ q^{8}$

$F_{9} = 2\ q + 2\ q^{2} + 28\ q^{3} + 34\ q^{4} + 130\ q^{5} + 102\ q^{6} + 144\ q^{7} + 54\ q^{8} + 16\ q^{9}$

Description

The length of the partition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

valleys-to-peaks

Description

Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence

$01$ , or a trailing

$0$ . A peak is a subsequence

$10$ or a trailing

$1$ . This map replaces every valley with a peak.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word

$w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in

$w$ .