Identifier
-
Mp00095:
Integer partitions
—to binary word⟶
Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000160: Integer partitions ⟶ ℤ
Values
[1] => 10 => [1,2] => [2,1] => 1
[2] => 100 => [1,3] => [3,1] => 1
[1,1] => 110 => [1,1,2] => [2,1,1] => 2
[3] => 1000 => [1,4] => [4,1] => 1
[2,1] => 1010 => [1,2,2] => [2,2,1] => 1
[1,1,1] => 1110 => [1,1,1,2] => [2,1,1,1] => 3
[4] => 10000 => [1,5] => [5,1] => 1
[3,1] => 10010 => [1,3,2] => [3,2,1] => 1
[2,2] => 1100 => [1,1,3] => [3,1,1] => 2
[2,1,1] => 10110 => [1,2,1,2] => [2,2,1,1] => 2
[1,1,1,1] => 11110 => [1,1,1,1,2] => [2,1,1,1,1] => 4
[5] => 100000 => [1,6] => [6,1] => 1
[4,1] => 100010 => [1,4,2] => [4,2,1] => 1
[3,2] => 10100 => [1,2,3] => [3,2,1] => 1
[3,1,1] => 100110 => [1,3,1,2] => [3,2,1,1] => 2
[2,2,1] => 11010 => [1,1,2,2] => [2,2,1,1] => 2
[2,1,1,1] => 101110 => [1,2,1,1,2] => [2,2,1,1,1] => 3
[1,1,1,1,1] => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1] => 5
[6] => 1000000 => [1,7] => [7,1] => 1
[5,1] => 1000010 => [1,5,2] => [5,2,1] => 1
[4,2] => 100100 => [1,3,3] => [3,3,1] => 1
[4,1,1] => 1000110 => [1,4,1,2] => [4,2,1,1] => 2
[3,3] => 11000 => [1,1,4] => [4,1,1] => 2
[3,2,1] => 101010 => [1,2,2,2] => [2,2,2,1] => 1
[3,1,1,1] => 1001110 => [1,3,1,1,2] => [3,2,1,1,1] => 3
[2,2,2] => 11100 => [1,1,1,3] => [3,1,1,1] => 3
[2,2,1,1] => 110110 => [1,1,2,1,2] => [2,2,1,1,1] => 3
[2,1,1,1,1] => 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1] => 4
[1,1,1,1,1,1] => 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 6
[7] => 10000000 => [1,8] => [8,1] => 1
[6,1] => 10000010 => [1,6,2] => [6,2,1] => 1
[5,2] => 1000100 => [1,4,3] => [4,3,1] => 1
[5,1,1] => 10000110 => [1,5,1,2] => [5,2,1,1] => 2
[4,3] => 101000 => [1,2,4] => [4,2,1] => 1
[4,2,1] => 1001010 => [1,3,2,2] => [3,2,2,1] => 1
[4,1,1,1] => 10001110 => [1,4,1,1,2] => [4,2,1,1,1] => 3
[3,3,1] => 110010 => [1,1,3,2] => [3,2,1,1] => 2
[3,2,2] => 101100 => [1,2,1,3] => [3,2,1,1] => 2
[3,2,1,1] => 1010110 => [1,2,2,1,2] => [2,2,2,1,1] => 2
[3,1,1,1,1] => 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1] => 4
[2,2,2,1] => 111010 => [1,1,1,2,2] => [2,2,1,1,1] => 3
[2,2,1,1,1] => 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1] => 4
[2,1,1,1,1,1] => 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1] => 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 7
[8] => 100000000 => [1,9] => [9,1] => 1
[7,1] => 100000010 => [1,7,2] => [7,2,1] => 1
[6,2] => 10000100 => [1,5,3] => [5,3,1] => 1
[6,1,1] => 100000110 => [1,6,1,2] => [6,2,1,1] => 2
[5,3] => 1001000 => [1,3,4] => [4,3,1] => 1
[5,2,1] => 10001010 => [1,4,2,2] => [4,2,2,1] => 1
[5,1,1,1] => 100001110 => [1,5,1,1,2] => [5,2,1,1,1] => 3
[4,4] => 110000 => [1,1,5] => [5,1,1] => 2
[4,3,1] => 1010010 => [1,2,3,2] => [3,2,2,1] => 1
[4,2,2] => 1001100 => [1,3,1,3] => [3,3,1,1] => 2
[4,2,1,1] => 10010110 => [1,3,2,1,2] => [3,2,2,1,1] => 2
[4,1,1,1,1] => 100011110 => [1,4,1,1,1,2] => [4,2,1,1,1,1] => 4
[3,3,2] => 110100 => [1,1,2,3] => [3,2,1,1] => 2
[3,3,1,1] => 1100110 => [1,1,3,1,2] => [3,2,1,1,1] => 3
[3,2,2,1] => 1011010 => [1,2,1,2,2] => [2,2,2,1,1] => 2
[3,2,1,1,1] => 10101110 => [1,2,2,1,1,2] => [2,2,2,1,1,1] => 3
[3,1,1,1,1,1] => 100111110 => [1,3,1,1,1,1,2] => [3,2,1,1,1,1,1] => 5
[2,2,2,2] => 111100 => [1,1,1,1,3] => [3,1,1,1,1] => 4
[2,2,2,1,1] => 1110110 => [1,1,1,2,1,2] => [2,2,1,1,1,1] => 4
[2,2,1,1,1,1] => 11011110 => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1] => 5
[2,1,1,1,1,1,1] => 101111110 => [1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1] => 111111110 => [1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 8
[9] => 1000000000 => [1,10] => [10,1] => 1
[8,1] => 1000000010 => [1,8,2] => [8,2,1] => 1
[7,2] => 100000100 => [1,6,3] => [6,3,1] => 1
[7,1,1] => 1000000110 => [1,7,1,2] => [7,2,1,1] => 2
[6,3] => 10001000 => [1,4,4] => [4,4,1] => 1
[6,2,1] => 100001010 => [1,5,2,2] => [5,2,2,1] => 1
[6,1,1,1] => 1000001110 => [1,6,1,1,2] => [6,2,1,1,1] => 3
[5,4] => 1010000 => [1,2,5] => [5,2,1] => 1
[5,3,1] => 10010010 => [1,3,3,2] => [3,3,2,1] => 1
[5,2,2] => 10001100 => [1,4,1,3] => [4,3,1,1] => 2
[5,2,1,1] => 100010110 => [1,4,2,1,2] => [4,2,2,1,1] => 2
[5,1,1,1,1] => 1000011110 => [1,5,1,1,1,2] => [5,2,1,1,1,1] => 4
[4,4,1] => 1100010 => [1,1,4,2] => [4,2,1,1] => 2
[4,3,2] => 1010100 => [1,2,2,3] => [3,2,2,1] => 1
[4,3,1,1] => 10100110 => [1,2,3,1,2] => [3,2,2,1,1] => 2
[4,2,2,1] => 10011010 => [1,3,1,2,2] => [3,2,2,1,1] => 2
[4,2,1,1,1] => 100101110 => [1,3,2,1,1,2] => [3,2,2,1,1,1] => 3
[4,1,1,1,1,1] => 1000111110 => [1,4,1,1,1,1,2] => [4,2,1,1,1,1,1] => 5
[3,3,3] => 111000 => [1,1,1,4] => [4,1,1,1] => 3
[3,3,2,1] => 1101010 => [1,1,2,2,2] => [2,2,2,1,1] => 2
[3,3,1,1,1] => 11001110 => [1,1,3,1,1,2] => [3,2,1,1,1,1] => 4
[3,2,2,2] => 1011100 => [1,2,1,1,3] => [3,2,1,1,1] => 3
[3,2,2,1,1] => 10110110 => [1,2,1,2,1,2] => [2,2,2,1,1,1] => 3
[3,2,1,1,1,1] => 101011110 => [1,2,2,1,1,1,2] => [2,2,2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => 1001111110 => [1,3,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1] => 6
[2,2,2,2,1] => 1111010 => [1,1,1,1,2,2] => [2,2,1,1,1,1] => 4
[2,2,2,1,1,1] => 11101110 => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1] => 5
[2,2,1,1,1,1,1] => 110111110 => [1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 6
[2,1,1,1,1,1,1,1] => 1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1,1] => 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1] => 9
[10] => 10000000000 => [1,11] => [11,1] => 1
[9,1] => 10000000010 => [1,9,2] => [9,2,1] => 1
[8,2] => 1000000100 => [1,7,3] => [7,3,1] => 1
[8,1,1] => 10000000110 => [1,8,1,2] => [8,2,1,1] => 2
[7,3] => 100001000 => [1,5,4] => [5,4,1] => 1
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Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
to partition
Description
Sends a composition to the partition obtained by sorting the entries.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
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