§26.9 Integer Partitions: Restricted Number and Part Size

Keywords:: partitions
Referenced by:: §17.16, §27.14(vi)
Permalink:: http://dlmf.nist.gov/26.9

§26.9(i) Definitions
§26.9(ii) Generating Functions
§26.9(iii) Recurrence Relations
§26.9(iv) Limiting Form

§26.9(i) Definitions

Notes:: See Andrews (1976, pp. 1–13, 81). Table 26.9.1 was computed by the author.
Keywords:: Ferrers graph, partitions, restricted integer partitions
Referenced by:: §26.17, §27.14(i), §27.14(i)
Permalink:: http://dlmf.nist.gov/26.9.i

$\mathop{p_{{k}}\/}\nolimits\!\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1.

26.9.1 $\mathop{p_{{k}}\/}\nolimits\!\left(n\right)=\mathop{p\/}\nolimits\!\left(n\right),$ $k\geq n$ .

Symbols:: $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.9.E1
Encodings:: TeX, pMML, png

Unrestricted partitions are covered in §27.14.

Table 26.9.1: Partitions $\mathop{p_{{k}}\/}\nolimits\!\left(n\right)$ .

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1	1	1	1	1	1	1	1	1	1	1
1	0	1	1	1	1	1	1	1	1	1	1
2	0	1	2	2	2	2	2	2	2	2	2
3	0	1	2	3	3	3	3	3	3	3	3
4	0	1	3	4	5	5	5	5	5	5	5
5	0	1	3	5	6	7	7	7	7	7	7
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
8	0	1	5	10	15	18	20	21	22	22	22
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

Symbols:: $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: restricted integer partitions
Referenced by:: §26.9(i), §26.9(i)
Permalink:: http://dlmf.nist.gov/26.9.T1

A useful representation for a partition is the Ferrers graph in which the integers in the partition are each represented by a row of dots. An example is provided in Figure 26.9.1.

$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$	$\bullet$
$\bullet$	$\bullet$	$\bullet$	$\bullet$
$\bullet$	$\bullet$	$\bullet$
$\bullet$	$\bullet$	$\bullet$
$\bullet$	$\bullet$
$\bullet$

Figure 26.9.1: Ferrers graph of the partition $7+4+3+3+2+1$ .

Referenced by:: §26.9(i), §26.9(i)
Permalink:: http://dlmf.nist.gov/26.9.F1

The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. The conjugate to the example in Figure 26.9.1 is $6+5+4+2+1+1+1$ . Conjugation establishes a one-to-one correspondence between partitions of $n$ into at most $k$ parts and partitions of $n$ into parts with largest part less than or equal to $k$ . It follows that $\mathop{p_{{k}}\/}\nolimits\!\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$ .

$\mathop{p_{{k}}\/}\nolimits\!\left(\leq m,n\right)$ is the number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ . It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$ , $1\leq h\leq m$ , $1\leq j\leq k$ , above and to the left of the lattice path. See Figure 26.9.2.

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Figure 26.9.2: The partition $5+5+3+2$ represented as a lattice path.

Symbols:: $(a,b)$ : open interval
Referenced by:: ¶ ‣ §26.2, §26.9(i)
Permalink:: http://dlmf.nist.gov/26.9.F2
Encodings:: TeX, png

Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer $k$ . See Andrews (1976, p. 81).

26.9.2 $\mathop{p_{{0}}\/}\nolimits\!\left(n\right)=0,$ $n>0$ ,

Symbols:: $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ and $n$ : nonnegative integer
Referenced by:: §26.9(i)
Permalink:: http://dlmf.nist.gov/26.9.E2
Encodings:: TeX, pMML, png

26.9.3

$\mathop{p_{{1}}\/}\nolimits\!\left(n\right)=1,$

$\mathop{p_{{2}}\/}\nolimits\!\left(n\right)=1+\left\lfloor n/2\right\rfloor,$

$\mathop{p_{{3}}\/}\nolimits\!\left(n\right)=1+\left\lfloor\frac{n^{2}+6n}{12}\right\rfloor.$

Symbols:: $\left\lfloor x\right\rfloor$ : floor of $x$ , $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ and $n$ : nonnegative integer
Referenced by:: §26.9(i)
Permalink:: http://dlmf.nist.gov/26.9.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

§26.9(ii) Generating Functions

Notes:: See Andrews (1976, pp. 36, 47).
Keywords:: Gaussian polynomials, $q$ -binomial coefficient, restricted integer partitions
Referenced by:: §26.16
Permalink:: http://dlmf.nist.gov/26.9.ii

In what follows

26.9.4 $\genfrac{[}{]}{0.0pt}{}{m}{n}_{{q}}=\prod _{{j=1}}^{n}\frac{1-q^{{m-n+j}}}{1-q^{j}},$ $n\geq 0$ ,

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E4
Encodings:: TeX, pMML, png

is the Gaussian polynomial (or $q$ -binomial coefficient); compare §§17.2(i)–17.2(ii). In the present chapter $m\geq n\geq 0$ in all cases. It is also assumed everywhere that $|q|<1$ .

26.9.5 $\sum _{{n=0}}^{{\infty}}\mathop{p_{{k}}\/}\nolimits\!\left(n\right)q^{n}=\prod _{{j=1}}^{k}\frac{1}{1-q^{j}}=1+\sum _{{m=1}}^{{\infty}}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{{q}}q^{m},$

Symbols:: $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ , $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E5
Encodings:: TeX, pMML, png

26.9.6 $\sum _{{n=0}}^{{\infty}}\mathop{p_{{k}}\/}\nolimits\!\left(\leq m,n\right)q^{n}=\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{{q}}.$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $\mathop{p\/}\nolimits\!\left(\mathrm{condition},n\right)$ : restricted number of partions of $n$ , $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E6
Encodings:: TeX, pMML, png

Also, when $|xq|<1$

26.9.7 $\sum _{{m,n=0}}^{{\infty}}\mathop{p_{{k}}\/}\nolimits\!\left(\leq m,n\right)x^{k}q^{n}=1+\sum _{{k=1}}^{{\infty}}\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{{q}}x^{k}=\prod _{{j=0}}^{m}\frac{1}{1-x\, q^{j}}.$

Symbols:: $\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $\mathop{p\/}\nolimits\!\left(\mathrm{condition},n\right)$ : restricted number of partions of $n$ , $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.9.E7
Encodings:: TeX, pMML, png

§26.9(iii) Recurrence Relations

Notes:: (26.9.9) is a special case of equation (2.23) in Bressoud (1999, p. 60).
Keywords:: restricted integer partitions
Permalink:: http://dlmf.nist.gov/26.9.iii

26.9.8 $\mathop{p_{{k}}\/}\nolimits\!\left(n\right)=\mathop{p_{{k}}\/}\nolimits\!\left(n-k\right)+\mathop{p_{{k-1}}\/}\nolimits\!\left(n\right);$

Symbols:: $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ , $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: §26.10(iii)
Permalink:: http://dlmf.nist.gov/26.9.E8
Encodings:: TeX, pMML, png

equivalently, partitions into at most $k$ parts either have exactly $k$ parts, in which case we can subtract one from each part, or they have strictly fewer than $k$ parts.

26.9.9 $\mathop{p_{{k}}\/}\nolimits\!\left(n\right)=\frac{1}{n}\sum _{{t=1}}^{n}\mathop{p_{{k}}\/}\nolimits\!\left(n-t\right)\sum _{{\substack{j\divides t\\ j\leq k}}}j,$

Symbols:: $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ , $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: §26.9(iii)
Permalink:: http://dlmf.nist.gov/26.9.E9
Encodings:: TeX, pMML, png

where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$ .

§26.9(iv) Limiting Form

Notes:: See Andrews (1976, Chapter 6).
Keywords:: restricted integer partitions
Permalink:: http://dlmf.nist.gov/26.9.iv

As $n\to\infty$ with $k$ fixed,

26.9.10 $\mathop{p_{{k}}\/}\nolimits\!\left(n\right)\sim\frac{n^{{k-1}}}{k!(k-1)!}.$

Symbols:: $!$ : $n!$ : factorial, $\mathop{p\/}\nolimits\!\left(n\right)$ : total number of partitions of $n$ , $\sim$ : asymptotic equality, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.9.E10
Encodings:: TeX, pMML, png

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1	1	1	1	1	1	1	1	1	1	1
1	0	1	1	1	1	1	1	1	1	1	1
2	0	1	2	2	2	2	2	2	2	2	2
3	0	1	2	3	3	3	3	3	3	3	3
4	0	1	3	4	5	5	5	5	5	5	5
5	0	1	3	5	6	7	7	7	7	7	7
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
8	0	1	5	10	15	18	20	21	22	22	22
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1	1	1	1	1	1	1	1	1	1	1
1	0	1	1	1	1	1	1	1	1	1	1
2	0	1	2	2	2	2	2	2	2	2	2
3	0	1	2	3	3	3	3	3	3	3	3
4	0	1	3	4	5	5	5	5	5	5	5
5	0	1	3	5	6	7	7	7	7	7	7
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
8	0	1	5	10	15	18	20	21	22	22	22
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

§26.9 Integer Partitions: Restricted Number and Part Size

Contents

§26.9(i) Definitions

§26.9(ii) Generating Functions

§26.9(iii) Recurrence Relations

§26.9(iv) Limiting Form

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1	1	1	1	1	1	1	1	1	1	1
1	0	1	1	1	1	1	1	1	1	1	1
2	0	1	2	2	2	2	2	2	2	2	2
3	0	1	2	3	3	3	3	3	3	3	3
4	0	1	3	4	5	5	5	5	5	5	5
5	0	1	3	5	6	7	7	7	7	7	7
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
8	0	1	5	10	15	18	20	21	22	22	22
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42