- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000175: Integer partitions ⟶ ℤ

Values

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

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

00 => [2] => [2] => 0

01 => [1,1] => [1,1] => 0

10 => [1,1] => [1,1] => 0

11 => [2] => [2] => 0

000 => [3] => [3] => 0

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

010 => [1,1,1] => [1,1,1] => 0

011 => [1,2] => [2,1] => 1

100 => [1,2] => [2,1] => 1

101 => [1,1,1] => [1,1,1] => 0

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

111 => [3] => [3] => 0

0000 => [4] => [4] => 0

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

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

0011 => [2,2] => [2,2] => 0

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

0101 => [1,1,1,1] => [1,1,1,1] => 0

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

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

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

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

1010 => [1,1,1,1] => [1,1,1,1] => 0

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

1100 => [2,2] => [2,2] => 0

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

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

1111 => [4] => [4] => 0

00000 => [5] => [5] => 0

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

00010 => [3,1,1] => [3,1,1] => 2

00011 => [3,2] => [3,2] => 1

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

00101 => [2,1,1,1] => [2,1,1,1] => 3

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

00111 => [2,3] => [3,2] => 1

01000 => [1,1,3] => [3,1,1] => 2

01001 => [1,1,2,1] => [2,1,1,1] => 3

01010 => [1,1,1,1,1] => [1,1,1,1,1] => 0

01011 => [1,1,1,2] => [2,1,1,1] => 3

01100 => [1,2,2] => [2,2,1] => 2

01101 => [1,2,1,1] => [2,1,1,1] => 3

01110 => [1,3,1] => [3,1,1] => 2

01111 => [1,4] => [4,1] => 1

10000 => [1,4] => [4,1] => 1

10001 => [1,3,1] => [3,1,1] => 2

10010 => [1,2,1,1] => [2,1,1,1] => 3

10011 => [1,2,2] => [2,2,1] => 2

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

10101 => [1,1,1,1,1] => [1,1,1,1,1] => 0

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

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

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

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

11010 => [2,1,1,1] => [2,1,1,1] => 3

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

11100 => [3,2] => [3,2] => 1

11101 => [3,1,1] => [3,1,1] => 2

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

11111 => [5] => [5] => 0

000000 => [6] => [6] => 0

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

000010 => [4,1,1] => [4,1,1] => 2

000011 => [4,2] => [4,2] => 1

000100 => [3,1,2] => [3,2,1] => 3

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

000110 => [3,2,1] => [3,2,1] => 3

000111 => [3,3] => [3,3] => 0

001000 => [2,1,3] => [3,2,1] => 3

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

001010 => [2,1,1,1,1] => [2,1,1,1,1] => 4

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

001100 => [2,2,2] => [2,2,2] => 0

001101 => [2,2,1,1] => [2,2,1,1] => 4

001110 => [2,3,1] => [3,2,1] => 3

001111 => [2,4] => [4,2] => 1

010000 => [1,1,4] => [4,1,1] => 2

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

010010 => [1,1,2,1,1] => [2,1,1,1,1] => 4

010011 => [1,1,2,2] => [2,2,1,1] => 4

010100 => [1,1,1,1,2] => [2,1,1,1,1] => 4

010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0

010110 => [1,1,1,2,1] => [2,1,1,1,1] => 4

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

011000 => [1,2,3] => [3,2,1] => 3

011001 => [1,2,2,1] => [2,2,1,1] => 4

011010 => [1,2,1,1,1] => [2,1,1,1,1] => 4

011011 => [1,2,1,2] => [2,2,1,1] => 4

011100 => [1,3,2] => [3,2,1] => 3

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

011110 => [1,4,1] => [4,1,1] => 2

011111 => [1,5] => [5,1] => 1

100000 => [1,5] => [5,1] => 1

100001 => [1,4,1] => [4,1,1] => 2

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

100011 => [1,3,2] => [3,2,1] => 3

100100 => [1,2,1,2] => [2,2,1,1] => 4

100101 => [1,2,1,1,1] => [2,1,1,1,1] => 4

100110 => [1,2,2,1] => [2,2,1,1] => 4

>>> Load all 2416 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$

Map

to partition

Description

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

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 $w$.