Partition function

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The partition function

p (n)

gives the number of partitions of a nonnegative integer

n

into positive integers. (There is one partition of zero into positive integers, i.e. the empty partition, since the empty sum is defined as 0.)

Table of partition function approximations

Note that

p (n) = F (n + 1), 0   ≤   n   ≤   4

, where

F (n)

is the

n

th Fibonacci number. For the Hardy–Ramanujan asymptotic approximation,

[·]

is the nearest integer function.

Partition function approximations

n	p (n), n   ≥   0 .	Hardy–Ramanujan asymptotic approximation HR (n) = 1 4 n 2√  3 exp  π 2√   2 n 3	HR (n)  −  p (n), n   ≥   1 .
	A000041 ​(n), n   ≥   0 .	A190840  (n), n   ≥   1 .		A035949  (n  −  3), n   ≥   4 .
< 0	0
0	1
1	1	2	1
2	2	3	1
3	3	4	1
4	5	6	1	0
5	7	9	2	1
6	11	13	2	1
7	15	18	3	2
8	22	26	4	2
9	30	35	5	4
10	42	48	6	4
11	56	65	9	7
12	77	87	10	8
13	101	115	14	12
14	135	152	17	14
15	176	199	23	20
16	231	258	27	23
17	297	333	36	32
18	385	427	42	39
19	490	545	55	51
20	627	692	65	61
21	792	875	83	80
22	1002	1102	100	95
23	1255	1381	126	122
24	1575	1725	150	146
25	1958	2145	187	183
26	2436	2659	223	219
27	3010	3285	275	273
28	3718	4046	328	324
29	4565	4967	402	399
30	5604	6080	476	475
31	6842	7423	581	578
32	8349	9037	688	685
33	10143	10974	831	830
34	12310	13293	983	979
35	14883	16065	1182	1177
36	17977	19370	1393	1387
37	21637	23304	1667	1655
38	26015	27977	1962	1945
39	31185	33519	2334	2311
40	37338	40080	2742	2705
41	44583	47833	3250	3198
42	53174	56981	3807	3737
43	63261	67757	4496	4396
44	75175	80431	5256	5121
45	89134	95316	6182	6003
46	105558	112771	7213	6973
47	124754	133211	8457	8143
48	147273	157115	9842	9439
49	173525	185031	11506	10981
50	204226	217590	13364	12697

The following table (compiled in OpenOffice.org Calc, wich has 15 digits max precision, implying that the last digit or two of 14 or 15 digits numbers is not reliable) compares two asymptotic approximations of

p (n)

, where the first one is the Hardy–Ramanujan asymptotic approximation. (I got the ROUND( EXP(PI()*SQRT(2*n/3)) / (4*(n+(1/2)*(LOG(n))^3)*SQRT(3)) ) (also asymptotic) approximation empirically. If both asymptotic approximations are always overestimates of

p (n)

, then HR2(n) is always a better approximation than HR(n), since HR2(n) is always less than HR(n). Thus arises the question: are HR(n) and HR2(n) always overstimates of

p (n)

? — Daniel Forgues 20:39, 31 July 2011 (UTC))

  n	               p(n)	              HR(n)	     HR(n) – p(n)	             HR2(n)	  HR2(n) – p(n)     (HR2(n) – p(n))
	            A000041 	    Hardy-Ramanujan                      (Modified) Hardy-Ramanujan                                       /
                                         asymptotic                               (also) asymptotic                          (HR(n) – p(n))
                              approximation of p(n)                           approximation of p(n)

                       ROUND( EXP(PI()*SQRT(2*n/3))                    ROUND( EXP(PI()*SQRT(2*n/3))
                                                  /                                               /
                                  (4*(n)*SQRT(3)) )              (4*(n+(1/2)*(LOG(n))^3)*SQRT(3)) )

  5	                  7	                  9	                2	                  9	              2                   1       
 10	                 42	                 48	                6	                 46	              4
 15	                176	                199	               23	                188	             12
 20	                627	                692	               65	                656	             29
 25	              1,958	              2,145	              187	              2,034	             76     0.4064171122994
 30	              5,604	              6,080	              476	              5,770	            166
 35	             14,883	             16,065	            1,182	             15,262	            379
 40	             37,338	             40,080	            2,742	             38,121	            783
 45	             89,134	             95,316	            6,182	             90,759	          1,625
 50	            204,226	            217,590	           13,364	            207,419	          3,193     0.2389254714157
 55	            451,276	            479,431	           28,155	            457,506	          6,230
 60	            966,467	          1,024,004	           57,537	            978,175	         11,708
 65	          2,012,558	          2,127,604	          115,046	          2,034,361	         21,803
 70	          4,087,968	          4,312,670	          224,702	          4,127,481	         39,513
 75	          8,118,264	          8,549,010	          430,746	          8,189,102	         70,838     0.1644542259243
 80	         15,796,476	         16,606,782	          810,306	         15,920,936	        124,460
 85	         30,167,357	         31,667,409	        1,500,052	         30,383,686	        216,329
 90	         56,634,173	         59,367,760	        2,733,587	         57,004,184	        370,011
 95	        104,651,419	        109,564,225	        4,912,806	        105,277,945	        626,526
100	        190,569,292	        199,280,893	        8,711,601	        191,616,244	      1,046,952     0.1201790577874
105	        342,325,709	        357,586,938	       15,261,229	        344,058,910	      1,733,201
110	        607,163,746	        633,589,186	       26,425,440	        610,001,367	      2,837,621
115	      1,064,144,451	      1,109,412,780	       45,268,329	      1,068,750,525	      4,606,074
120	      1,844,349,560	      1,921,105,572	       76,756,012	      1,851,755,110	      7,405,550
125	      3,163,127,352	      3,292,035,176	      128,907,824	      3,174,941,068	     11,813,716     0.0916446778280
130	      5,371,315,400	      5,585,841,896	      214,526,496	      5,390,004,417	     18,689,017
135	      9,035,836,076	      9,389,797,022	      353,960,946	      9,065,191,305	     29,355,229
140	     15,065,878,135	     15,645,130,511	      579,252,376	     15,111,647,413	     45,769,278
145	     24,908,858,009	     25,849,469,513	      940,611,504	     24,979,755,750	     70,897,741
150	     40,853,235,313	     42,369,336,269	    1,516,100,956	     40,962,334,140	    109,098,827     0.0719601333725
155	     66,493,182,097	     68,919,680,135	    2,426,498,038	     66,660,068,881	    166,886,784
160	    107,438,159,466	    111,295,547,185	    3,857,387,719	    107,691,928,166	    253,768,700
165	    172,389,800,255	    178,482,384,132	    6,092,583,877	    172,773,583,796	    383,783,541
170	    274,768,617,130	    284,332,114,132	    9,563,497,002	    275,345,904,405	    577,287,275
175	    435,157,697,830	    450,080,578,578	   14,922,880,748	    436,021,721,915	    864,024,085     0.0578992822894
180	    684,957,390,936	    708,110,435,576	   23,153,044,640	    686,244,237,116	  1,286,846,180
185	  1,071,823,774,337	  1,107,549,520,159	   35,725,745,822	  1,073,731,572,223	  1,907,797,886
190	  1,667,727,404,093	  1,722,562,621,633	   54,835,217,540	  1,670,543,118,819	  2,815,714,726
195	  2,580,840,212,973	  2,664,579,377,303	   83,739,164,330	  2,584,978,391,028	  4,138,178,055
200	  3,972,999,029,388	  4,100,251,432,188	  127,252,402,800	  3,979,055,824,394	  6,056,795,006     0.0475967044451
205	  6,085,253,859,260	  6,277,716,787,260	  192,462,928,000	  6,094,084,357,824	  8,830,498,564
210	  9,275,102,575,355	  9,564,864,314,565	  289,761,739,210	  9,287,928,415,566	 12,825,840,211
215	 14,070,545,699,287	 14,504,870,558,863	  434,324,859,576	 14,089,107,769,612	 18,562,070,325
220	 21,248,279,009,367	 21,896,510,200,516	  648,231,191,149	 21,275,049,463,813	 26,770,454,446
225	 31,946,390,696,157	 32,909,878,835,929	  963,488,139,772	 31,984,871,555,578	 38,480,859,421     0.0399391106465
230	 47,826,239,745,920	 49,252,568,636,892	1,426,328,890,972	 47,881,376,419,261	 55,136,673,341
235	 71,304,185,514,919	 73,407,495,709,204	2,103,310,194,285	 71,382,945,468,515	 78,759,953,596
240	105,882,246,722,733	108,972,168,902,833	3,089,922,180,100	105,994,418,678,409	112,171,955,676
245	156,618,412,527,946	161,141,141,284,870	4,522,728,756,924	156,777,718,615,992	159,306,088,046
250	230,793,554,364,681	237,389,967,288,174	6,596,412,923,493	231,019,181,853,355	225,627,488,674     0.0342045731962

Table of related formulae and values

Partition function related formulae and values

n	p (n) A000041	Differences p (n)  −  p (n  −  1) = p (2, n), n   ≥   1 A002865	Partial sums   m ∑   n  = 0    p (n) A000070	Partial sums of reciprocals   m ∑   n  = 0    1 p (n)
< 0	0	0	0
0	1	1	1
1	1	0	2
2	2	1	4
3	3	1	7
4	5	2	12
5	7	2	19
6	11	4	30
7	15	4	45
8	22	7	67
9	30	8	97
10	42	12	139
11	56	14	195
12	77	21	272
13	101	24	373
14	135	34	508
15	176	41	684
16	231	55	915
17	297	66	1212
18	385	88	1597
19	490	105	2087
20	627	137	2714
21	792	165	3506
22	1002	210	4508
23	1255	253	5763
24	1575	320	7338
25	1958	383	9296
26	2436	478	11732
27	3010	574	14742
28	3718	708	18460
29	4565	847	23025
30	5604	1039	28629
31	6842	1238	35471
32	8349	1507	43820
33	10143	1794	53963
34	12310	2167	66273
35	14883	2573	81156
36	17977	3094	99133
37	21637	3660	120770
38	26015	4378	146785
39	31185	5170	177970
40	37338	6153	215308
41	44583	7245	259891
42	53174	8591	313065
43	63261	10087	376326
44	75175	11914	451501
45	89134	13959	540635
46	105558	16424	646193
47	124754	19196	770947
48	147273	22519	918220
49	173525	26252	1091745
50	204226	30701	1295971

Generating function of the partition function

The generating function of

p (n)

is the reciprocal of Euler’s function

$G_{\{p(n)\}}(x)=\sum _{n=0}^{\infty }p(n)x^{n}=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{k}}}\right)={\frac {1}{\prod _{k=1}^{\infty }(1-x^{k})}}.$

Asymptotic behavior of the partition function

An asymptotic formula, first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920, for

p (n)

is given by

$p(n)\sim {\frac {1}{4n{\sqrt {3}}}}~e^{\pi {\sqrt {\frac {2n}{3}}}}={\frac {1}{8n\sin({\frac {\pi }{3}})}}~e^{{\frac {\pi }{3}}\sin({\frac {\pi }{3}}){\sqrt {8n}}}\,$

where

e π

is Gelfond’s constant.

Partition function formula

The partition function

p (n)

, is given by

$p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n)~k^{\frac {1}{2}}{\Bigg [}{\frac {d}{dx}}{\frac {\sinh {\bigg (}{\frac {\pi }{k}}{\big (}{\frac {2}{3}}{\big )}^{\frac {1}{2}}{\big (}x-{\frac {1}{24}}{\big )}^{\frac {1}{2}}{\bigg )}}{{\big (}x-{\frac {1}{24}}{\big )}^{\frac {1}{2}}}}{\Bigg ]}_{x=n},\,$

where

Ak (n)

is an explicitly given finite sum of 24th roots of unity. Note that

p (n) = 0

when

n < 0

since a negative integer is not the sum of positive integers, and

p (0) = 1

since the empty partition gives the empty sum, defined as the additive identity, i.e. 0.

Partition function recurrence

Euler’s pentagonal number theorem begets a finite (since

p (n) = 0

for

n < 0

) recursive formula for the partition function

$p(n)=\sum _{j=1}^{\infty }(-1)^{j-1}[p(n-q_{j})+p(n-q_{-j})]=p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-12)+p(n-15)-\cdots \,$

where

qj = j (3j  −  1) 2 , j ∈ ℤ,

are the generalized pentagonal numbers, with

j

going through the integer sequence (A001057)

{0, 1, −1, 2, −2, 3, −3, 4, − 4, 5, −5, 6, − 6, 7, −7, 8, −8, 9, − 9, 10, −10, 11, −11, 12, −12, 13, −13, 14, −14, 15, −15, 16, −16, 17, −17, 18, −18, 19, −19, 20, −20, 21, −21, 22, −22, 23, −23, 24, −24, 25, −25, 26, −26, ...}

giving the sequence of generalized pentagonal numbers

qj

(A001318)

{0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, ...}

The recurrence may be expressed with explicit finite bounds for the summation indexes

j

$p(n)=\sum _{j=1}^{{\big \lfloor }{\tfrac {1+{\sqrt {1+24n}}}{6}}{\big \rfloor }}(-1)^{j-1}~p{\big (}n-{\tfrac {j(3j-1)}{2}}{\big )}+\sum _{j=1}^{{\big \lfloor }{\tfrac {-1+{\sqrt {1+24n}}}{6}}{\big \rfloor }}(-1)^{j-1}~p{\big (}n-{\tfrac {j(3j+1)}{2}}{\big )}\,$

Intermediate partition function

The number of partitions of

n

using only positive integers at least as large as

k

, called the intermediate partition function

p (k, n)

, is given by

$p(k,n)=?,\,$

where

p (k, n) = 0

when

n < 0

or

k > n > 0

, and

p (k, 0) = 1

since the empty partition gives the empty sum (the additive identity, 0).

Intermediate partition function formula

The intermediate partition function formula is

$p(n)=1+\sum _{i=1}^{\lfloor {\frac {n}{2}}\rfloor }p(i,n-i)\,$

where we have the empty sum (the additive identity, 0) for the summation when

n < 2

, thus giving

p (n) = 1, 0   ≤   n   ≤   1

.

Intermediate partition function formula (other version)

The partition function may be expressed in terms of the intermediate partition function trivially as

$p(n)=p(1,n),\,n\,\geq \,0,\,$

and since every partition of

n

with smallest part 1 is trivially obtained from the partition of

n  −  1

without that part 1, we have

$p(0)=1,\,p(n)=p(n-1)+p(2,n),\,n\,\geq \,1,\,$

and by recursion we get

$p(0)=1,\,p(n)=[p(n-2)+p(2,n-1)]+p(2,n),\,n\,\geq \,1,\,$

...

$p(0)=1,\,p(n)=[p(n-3)+p(2,n-2)]+p(2,n-1)+p(2,n),\,n\,\geq \,1,\,$

...

$p(0)=1,\,p(n)=p(0)+\sum _{i=1}^{n}p(2,i),\,n\,\geq \,1,\,$

which may be rewritten as (where

p (k, 0) = 1

since the empty partition gives the empty sum, i.e. the additive identity, 0)

$p(n)=\sum _{i=0}^{n}p(2,i),\,n\,\geq \,0,\,$

or

$p(n)=1+\sum _{i=1}^{n}p(2,i),\,n\,\geq \,0.\,$

Partition function finite formulas

Emory University mathematicians Ken Ono and Zachary Kent, together with Amanda Folsom of Yale, have discovered that partition numbers behave like fractals. Ono and Jan Hendrik Bruinier, of the Technical University of Darmstadt, have devised the first finite formula to calculate the partitions of any number.

The partition function can be expressed as a sum over the pentagonal partitions of

n

. Let

n = k1 + 2 k2 + 5 k5 + ⋯ = ∑ m m qm kqm

be a pentagonal partition of

n

, where each

qm = m (3m  −  1) 2

is a generalized pentagonal number (GPN), sequence A001318, with

qM

being the largest GPN less than or equal to

n

. Then

$p(n)=\sum _{k_{1}=0}^{n}\sum _{k_{2}=0}^{\lfloor n/2\rfloor }\sum _{k_{5}=0}^{\lfloor n/5\rfloor }\cdots \sum _{k_{q_{M}}=0}^{\lfloor n/q_{M}\rfloor }(-1)^{A}\left({\begin{array}{c}K\\k_{1},k_{2},k_{5},\ldots ,k_{q_{M}}\end{array}}\right)\delta _{n,\sum q_{m}\,k_{q_{m}}},$

where

${\begin{aligned}K&=k_{1}+k_{2}+k_{5}+k_{7}+\cdots ,\\A&=k_{5}+k_{7}+k_{22}+k_{26}+\cdots =\sum _{m=\pm 2,\pm 4,\ldots }k_{q_{m}},\end{aligned}}$

and where

$\left({\begin{array}{c}K\\k_{1},\ldots ,k_{n}\end{array}}\right)\equiv {\frac {K!}{k_{1}!\cdots k_{n}!}}~\delta _{K,k_{1}+\cdots +k_{n}}$

is a multinomial coefficient. The number of terms in the sum for

p (n)

is sequence A095699, e.g. , and

${\begin{aligned}p(8)&=-\left({\begin{array}{c}2\\1,0,0,1\end{array}}\right)-\left({\begin{array}{c}3\\1,1,1,0\end{array}}\right)-\left({\begin{array}{c}4\\3,0,1,0\end{array}}\right)+\left({\begin{array}{c}4\\0,4,0,0\end{array}}\right)\\&~~~~\,+\left({\begin{array}{c}5\\2,3,0,0\end{array}}\right)+\left({\begin{array}{c}6\\4,2,0,0\end{array}}\right)+\left({\begin{array}{c}7\\6,1,0,0\end{array}}\right)+\left({\begin{array}{c}8\\8,0,0,0\end{array}}\right)\\&=-2-6-4+1+10+15+7+1=22.\end{aligned}}$

Determinant formulas

The value of the partition function

p (n)

can be found from

${\begin{matrix}{\text{GPN's}}&~\\{\begin{matrix}0\\1\\2\\~\\~\\5\\~\\7\\~\\\vdots \\\end{matrix}}&p(n)={\begin{vmatrix}~1&-1&~\\~1&~1&-1&~\\~0&~1&~1&-1&~\\~0&~0&~1&~1&-1&~\\-1&~0&~0&~1&~1&-1&~\\~0&-1&~0&~0&~1&~1&-1&~\\-1&~0&-1&~0&~0&~1&~1&-1&~\\~0&-1&~0&-1&~0&~0&~1&~1&-1&~\\~0&~0&-1&~0&-1&~0&~0&~1&~1&~\\~\vdots &~&~&~&~&~&~&~&~&\ddots \\\end{vmatrix}}_{n\times n}\end{matrix}}.$

p (n)

is the determinant of the

n  ×  n

truncation of the infinite-dimensional Toeplitz matrix shown above. The only non-zero diagonals of this matrix are those which start on a row labeled by a generalized pentagonal number

qm

. (The superdiagonal is taken to start on row 0.) On these diagonals, the matrix element is

( − 1) m+1

.

By using the identity by Ramanujan,

$\sum _{k=0}^{\infty }p(5k+4)\,x^{k}=5\,{\frac {(x^{5})_{\infty }^{5}}{(x)_{\infty }^{6}}},\,(x)_{\infty }\equiv \prod _{m=1}^{\infty }(1-x^{m}),$

the partition function

p (5k  −  1)

can be expressed alternatively as the determinant of a lower,

k

-dimensional matrix:

$p(5k-1)=5\,{\begin{vmatrix}~1&~&~&~&~&~&~&~1~\\-6&~1&~&~&~&~&~&~0~\\~9&-6&~1&~&~&~&~&~0~\\~10&~9&-6&~1&~&~&~&~0~\\-30&~10&~9&-6&~1&~&~&~0~\\~0&-30&~10&~9&-6&~1&~&-5~\\~11&~0&-30&~10&~9&-6&~&~0~\\~\vdots &~&~&~&~&~&\ddots &~\vdots ~\\\end{vmatrix}}_{k\times k}.$

The numbers making up the 1^st column correspond to sequence A000729, while every 5^th element (after the initial 1) in the last column is from sequence A000728:

{1, −5, 5, 10, −15, − 6, ...}

; all other elements in that column are zero. For example,

$p(29)=5\,{\begin{vmatrix}~1&~&~&~&~&~1~\\-6&~1&~&~&~&~0~\\~9&-6&~1&~&~&~0~\\~10&~9&-6&~1&~&~0~\\-30&~10&~9&-6&~1&~0~\\0&-30&~10&~9&-6&-5~\\\end{vmatrix}}=4565.$

In a like fashion, by using two other identities by Ramanujan the partition functions

p (7k  −  2)

and

p (25k  −  1)

can also be expressed as

k

-dimensional determinants:

${\begin{aligned}p(7k-2)&=7\,{\begin{vmatrix}~1&~&~&~&1~\\-8&~1&~&~&3~\\~20&-8&~1&~&2~\\~0&~20&-8&~&8~\\~\vdots &~&~&\ddots &\vdots ~\\\end{vmatrix}}_{k\times k},\\p(25k-1)&=25\,{\begin{vmatrix}~1&~&~&~&63~\\-31&~1&~&~&4988~\\~434&-31&~1&~&95751~\\-3565&~434&-31&~&766014~\\~\vdots &~&~&\ddots &\vdots ~\\\end{vmatrix}}_{k\times k},\end{aligned}}$

where the first column in each matrix is sequence A000731 and A010836, respectively, and the last column in each is from the expansions:

${\begin{aligned}(x)_{\infty }^{4}(x^{7})_{\infty }^{3}+7x(x^{7})_{\infty }^{7}&=1+3x+2x^{2}+8x^{3}+\cdots ;\\&\\63(x)_{\infty }^{24}(x^{5})_{\infty }^{6}+5^{3}\cdot 52x(x)_{\infty }^{18}(x^{5})_{\infty }^{12}&+5^{5}\cdot 63x^{2}(x)_{\infty }^{12}(x^{5})_{\infty }^{18}\\+~5^{8}\cdot 6x^{3}(x)_{\infty }^{6}(x^{5})_{\infty }^{24}&+5^{10}\cdot x^{4}(x^{5})_{\infty }^{30}\\&=63+4988x+95751x^{2}+766014x^{3}+\cdots .\end{aligned}}$

As an example, the calculation of

p (149)

is as follows:

$p(149)=25\,{\begin{vmatrix}~1&~&~&~&~&~63~\\-31&~1&~&~&~&~4988~\\~434&-31&~1&~&~&~95751~\\-3565&~434&-31&~1&~&~766014~\\~18445&-3565&~434&-31&~1&~3323665~\\-57505&~18445&-3565&~434&-31&~8359848~\\\end{vmatrix}}=37027355200.$

The partition function for partitions into distinct integers,

q (n)

, is

$q(n)={\begin{vmatrix}~1&~&~&~&~&~&~&~&~1~\\-1&~1&~&~&~&~&~&~&~0~\\-1&-1&~1&~&~&~&~&~&-1~\\~0&-1&-1&~1&~&~&~&~&~0~\\~0&~0&-1&-1&~1&~&~&~&-1~\\~1&~0&~0&-1&-1&~1&~&~&~0~\\~0&~1&~0&~0&-1&-1&~1&~&~0~\\~1&~0&~1&~0&~0&-1&-1&~&~0~\\~\vdots &~&~&~&~&~&~&\ddots &~\vdots ~\\\end{vmatrix}}_{(n+1)\times (n+1)},$

where the first column is sequence A010815, and a matrix element in the last column equals

( − 1) m

if the row number is

2 qm + 1

; otherwise it equals zero. By a result due to Euler,

q (n)

is also the partition function for partitions into odd integers.

First differences of the partition function

The first differences of the partition function are

$p(n+1)-p(n)=p(2,n+1),\,n\geq 1,\,$

where $\scriptstyle p(2,n+1)\,$ is the number of partitions with parts of size 2 or larger.

Partial sums of the partition function

The n-th partial sum of the partition function is equal to the n X n deternminant

$\sum _{k=0}^{n}p(k)={\begin{vmatrix}~2&-1&~\\~0&~2&-1&~\\-1&~0&~2&-1&~\\~0&-1&~0&~2&-1&~\\-1&~0&-1&~0&~2&-1&~\\~1&-1&~0&-1&~0&~2&-1&~\\~\vdots &~&~&~&~&~&\ddots &\ddots \\c_{n-1}&c_{n-2}&~&~&\cdots &~&~&~2&-1\\c_{n}&c_{n-1}&~&~&\cdots &~&~&~0&~2\end{vmatrix}}_{n\times n},$

where c₁ = 2, c₂ = 0, and for k > 2,

$c_{k}=\left\{{\begin{aligned}(-1)^{m+1}&&{\text{ if }}k=q_{m};\\(-1)^{m}&&{\text{ if }}k=q_{m}+1;\\0&&{\text{otherwise}}.\end{aligned}}\right.$

Partial sums of reciprocals of the partition function

The partials sums of reciprocals of the partition function are

$\sum _{n=0}^{m}{\frac {1}{p(n)}}=~?\,$

Sum of reciprocals of the partition function

The sum of reciprocals of the partition function is

$\sum _{n=0}^{\infty }{\frac {1}{p(n)}}=~?\,$

The decimal expansion is $\scriptstyle 2.510597483886\dots \,$ (A078506)

Order of basis of representation of integers as sum of partition numbers

The order of basis of the representation of integers as sum of partition numbers is

$?\,$

Sequences

A000041 Number of partitions

p (n)

of

n, n   ≥   0

.

{1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, ...}

A002865 Number of partitions

p (2, n)

of

n, n   ≥   0,

that do not contain 1 as a part. Corresponds to

p (n)  −  p (n  −  1), n   ≥   0

.

{1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, ...}

A000070 Summatory function of the partition function,

∑ n k  = 0 p (k)

.

{1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, 684, 915, 1212, 1597, 2087, 2714, 3506, 4508, 5763, 7338, 9296, 11732, 14742, 18460, 23025, 28629, 35471, 43820, 53963, 66273, 81156, ...}

A078506 Decimal expansion of sum of inverses of unrestricted partition function.

{2, 5, 1, 0, 5, 9, 7, 4, 8, 3, 8, 8, 6, 2, 9, 3, 9, 5, 3, 2, 3, 6, 8, 3, 4, 7, 2, 7, 4, 1, 5, 4, 6, 5, 4, 5, 1, 6, 8, 3, 5, 3, 1, 9, 4, 4, 9, 5, 5, 1, 4, 7, 6, 8, 1, 9, 0, 8, 0, 6, 2, 9, 9, 6, 5, 0, 8, 3, 8, 4, 5, 3, 2, 9, 0, 4, 4, ...}

A190840

1 4n 2√  3 exp π 2√   2n 3

rounded to nearest integer.

{2, 3, 4, 6, 9, 13, 18, 26, 35, 48, 65, 87, 115, 152, 199, 258, 333, 427, 545, 692, 875, 1102, 1381, 1725, 2145, 2659, 3285, 4046, 4967, 6080, 7423, 9037, 10974, 13293, 16065, 19370, 23304, 27977, 33519, ...}

A035949

(n), n   ≥   1

. Number of partitions in parts not of the form

13 k, 13 k + 1

or

13 k  −  1

. Also number of partitions with no part of size 1 and differences between parts at distance 5 are greater than 1.

{0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 23, 32, 39, 51, 61, 80, 95, 122, 146, 183, 219, 273, 324, 399, 475, 578, 685, 830, 979, 1177, 1387, 1655, 1945, 2311, 2705, 3198, 3737, 4396, 5121, 6003, 6973, 8143, 9439, ...}

See also

• Partitions
• Partition identities
• Orderings of partitions

External links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Jan Hendrik Bruinier and Ken Ono, An Algebraic Formula for the Partition Function.
• Amanda Folsom, Zachary A. Kent, and Ken Ono, p-adic Properties of the Partition Function.
• Jerome Kelleher and Barry O’Sullivan, Generating All Partitions: A Comparison of Two Encodings, 2009.
• Peter Luschny, Counting with Partitions, 2009–02–20.
• Omar E. Pol, 3D image showing the partitions of 1 to 9.
• W. J. A. Colman, A General Method for Determining a Closed Formula for the Number of Partitions of the Integer n into m Positive Integers for Small Values of m, submitted May 1982.