Partitions

Seems like the columns tend to the Fibonacci series (1,1,2,3,5,8,13,21,34,55,…)
where one term is the sum of the two previous ones.

Quote: kubikulann

Quote: gordonm888

I note that one can assign a number of permutations to any partition (e.g.: the partition 3-2-1-1 has 12 permutations, ) and then sum up the number of permutations for every entry in this table. When I do that I get this table:

	1	2	3	4	5	6	7	8	9	10	Total
1	1										1
2	1	1									2
3	1	2	1								4
4	1	3	3	1							8
5	1	4	6	4	1						16
6	1	5	10	10	5	1					32
7	1	6	15	20	15	6	1				64
8	1	7	21	35	35	21	7	1			128
9	1	8	28	56	70	56	28	8	1		256
10	1	9	36	84	126	126	84	36	9	1	512

This is Pascal’s table. (aka Binomial coefficients table)

Lottery numerology
A combo of 6 numbers occurs 2 times: diagonally and vertically
3 6 10 15 21 28
They are birthday numbers. Many people swear by birth numbers and some won millions with them in lottos all over the world.
Im gonna play em. Thanks all.

Quote: Wizard

Sorry to interrupt a good discussion, but here is a count of the number of pieces by size. The left column shows the piece size and along the top row is the base number that is getting partitioned.

Piece	1	2	3	4	5	6	7	8	9	10
1	1	2	4	7	12	19	30	45	67	97
2		1	1	3	4	8	11	19	26	41
3			1	1	2	4	6	9	15	21
4				1	1	2	3	6	8	13
5					1	1	2	3	5	8
6						1	1	2	3	5
7							1	1	2	3
8								1	1	2
9									1	1
10										1
Total	1	3	6	12	20	35	54	86	128	192

Offhand, the only interesting pattern I'm seeing is the total number of pieces generally divide lots of ways:

6 = 2*3
12 = 2*2*3
20 =2*2*5
35 = 5*7
54 = 2*3*3*3
86 = 2 * 43
128 = 2*2*2*2*2*2*2
192 = 2*2*2*2*2*2*3

I guess 86 is the exception.

If you keep a running sum/total of the Partitions Numbers and add one to that running sum you get:
N...P(N)....(running Sum of P(N)) +1
1___1______2
2___2______4
3___3______7
4___5______12
5___7______19
6___11_____30
7___15_____45
8___22_____67
9___30_____97

Notice that the (running sum of the partition number +1) = number of pieces that are 1 !

Edit: Another way to say this: think of the "Pieces of 1" row as a sequence of numbers. If you take the nth term and subtract the (n-1) term, you obtain a sequence that is the partition numbers!!

When you look at primes, you can't find any patterns. When you look at partitions, everything is a recursive pattern! Everywhere you look!!! Its maddening!!!! It's TURTLES all the way down!

The long silence after my last post leaves me wondering whether Wizard (and others) are attempting to write an algorithm, using combination math, to calculate the number of "pieces of 1" in partitions of any number N - as a way to calculate the partition number of any number N. I've been trying to conceptually develop the algorithm in my head, to judge whether its worth actually trying to attempt it. I don't know - its uncertain of success.

Edit: I think that such an attempt is likely to wind up with an algorithm that is a "sum of a sum" of terms, amounting to a sum of N[sup}2 terms. That's essentially the same as developing a table of N[sup}2 entries, as we are doing now. Unless there is a way to represent many terms by a function (like defining an infinite series to be a sin() function) I question whether such an approach will work.

Quote: gordonm888

The long silence after my last post ...

Sorry I was out of town for a few days. I'll get caught up shortly. Just skimming this, I find it intriguing, but need to chew on it some more.

Let me put into a table, what you said, because I love tables. The third column represents the sum of partitions from 1 to n-1, for n. For examples, for n=5, it equals P(1)+ P(2) + P(3) + P(4). As Gordon said, this column equals one less than the number of pieces of size one in the partition of n.

n	Partitions	Sum from 1 to n-1	Size 1
1	1	0	1
2	2	1	2
3	3	3	4
4	5	6	7
5	7	11	12
6	11	18	19
7	15	29	30
8	22	44	45
9	30	66	67
10	42	96	97
11	56	138	139
12	77	194	195
13	101	271	272
14	135	372	373
15	176	507	508

Offhand, I can't think of why this is true.

I found a very good estimate for the P(n) if you know P(n-1).

Here it is

P(n) = P(n-1) * exp(0.923266 * n^-0.448858)

Total	Partitions	Estimate
2	2	2
3	3	4
4	5	5
5	7	8
6	11	11
7	15	16
8	22	22
9	30	31
10	42	42
11	56	58
12	77	76
13	101	103
14	135	134
15	176	178
16	231	230
17	297	299
18	385	382
19	490	492
20	627	623
21	792	793
22	1,002	997
23	1,255	1,256
24	1,575	1,567
25	1,958	1,958
26	2,436	2,425
27	3,010	3,006
28	3,718	3,702
29	4,565	4,558
30	5,604	5,579
31	6,842	6,829
32	8,349	8,314
33	10,143	10,118
34	12,310	12,261
35	14,883	14,844
36	17,977	17,904
37	21,637	21,578
38	26,015	25,914
39	31,185	31,093
40	37,338	37,197
41	44,583	44,450
42	53,174	52,975
43	63,261	63,069
44	75,175	74,902
45	89,134	88,857
46	105,558	105,184
47	124,754	124,368
48	147,273	146,758
49	173,525	172,989
50	204,226	203,529
51	239,943	239,201
52	281,589	280,650
53	329,931	328,922
54	386,155	384,890
55	451,276	449,910
56	526,823	525,136
57	614,154	612,313
58	715,220	712,982
59	831,820	829,364
60	966,467	963,500
61	1,121,505	1,118,247
62	1,300,156	1,296,257
63	1,505,499	1,501,189
64	1,741,630	1,736,523
65	2,012,558	2,006,902
66	2,323,520	2,316,851
67	2,679,689	2,672,297
68	3,087,735	3,079,074
69	3,554,345	3,544,718
70	4,087,968	4,076,762
71	4,697,205	4,684,738
72	5,392,783	5,378,324
73	6,185,689	6,169,606
74	7,089,500	7,070,934
75	8,118,264	8,097,582
76	9,289,091	9,265,325
77	10,619,863	10,593,389
78	12,132,164	12,101,831
79	13,848,650	13,814,875
80	15,796,476	15,757,907
81	18,004,327	17,961,369
82	20,506,255	20,457,367
83	23,338,469	23,284,033
84	26,543,660	26,481,857
85	30,167,357	30,098,590
86	34,262,962	34,185,095
87	38,887,673	38,801,042
88	44,108,109	44,010,271
89	49,995,925	49,887,144
90	56,634,173	56,511,563
91	64,112,359	63,976,143
92	72,533,807	72,380,601
93	82,010,177	81,840,049
94	92,669,720	92,478,779
95	104,651,419	104,439,541
96	118,114,304	117,876,907
97	133,230,930	132,967,732
98	150,198,136	149,903,764
99	169,229,875	168,903,715
100	190,569,292	190,205,143
101	214,481,126	214,077,983
102	241,265,379	240,815,950
103	271,248,950	270,751,831
104	304,801,365	304,248,019
105	342,325,709	341,714,098
106	384,276,336	383,596,583
107	431,149,389	430,398,690
108	483,502,844	482,669,611
109	541,946,240	541,026,864
110	607,163,746	606,144,673
111	679,903,203	678,779,658
112	761,002,156	759,758,436
113	851,376,628	850,006,583
114	952,050,665	950,535,914
115	1,064,144,451	1,062,477,315
116	1,188,908,248	1,187,067,276
117	1,327,710,076	1,325,685,559
118	1,482,074,143	1,479,841,222
119	1,653,668,665	1,651,215,227
120	1,844,349,560	1,841,646,583
121	2,056,148,051	2,053,180,746
122	2,291,320,912	2,288,055,476
123	2,552,338,241	2,548,756,438
124	2,841,940,500	2,838,003,205
125	3,163,127,352	3,158,812,305
126	3,519,222,692	3,514,484,355
127	3,913,864,295	3,908,675,862
128	4,351,078,600	4,345,387,189
129	4,835,271,870	4,829,045,048
130	5,371,315,400	5,364,492,060
131	5,964,539,504	5,957,080,654
132	6,620,830,889	6,612,665,660
133	7,346,629,512	7,337,711,517
134	8,149,040,695	8,139,287,915
135	9,035,836,076	9,025,193,137
136	10,015,581,680	10,003,954,084
137	11,097,645,016	11,084,967,033
138	12,292,341,831	12,278,504,303
139	13,610,949,895	13,595,875,397
140	15,065,878,135	15,049,440,854
141	16,670,689,208	16,652,797,825
142	18,440,293,320	18,420,803,387
143	20,390,982,757	20,369,787,021
144	22,540,654,445	22,517,586,846
145	24,908,858,009	24,883,793,414
146	27,517,052,599	27,489,800,523
147	30,388,671,978	30,359,086,343
148	33,549,419,497	33,517,282,606
149	37,027,355,200	36,992,497,240
150	40,853,235,313	40,815,407,361
151	45,060,624,582	45,019,630,316
152	49,686,288,421	49,641,843,916
153	54,770,336,324	54,722,214,741
154	60,356,673,280	60,304,551,933
155	66,493,182,097	66,436,799,884
156	73,232,243,759	73,171,234,091
157	80,630,964,769	80,565,028,699
158	88,751,778,802	88,680,500,984
159	97,662,728,555	97,585,767,147
160	107,438,159,466	107,355,045,666
161	118,159,068,427	118,069,412,967
162	129,913,904,637	129,817,179,016
163	142,798,995,930	142,694,759,153
164	156,919,475,295	156,807,133,501
165	172,389,800,255	172,268,855,018
166	189,334,822,579	189,204,609,387
167	207,890,420,102	207,750,378,180
168	228,204,732,751	228,054,120,182
169	250,438,925,115	250,277,114,645
170	274,768,617,130	274,594,783,747
171	301,384,802,048	301,198,246,569
172	330,495,499,613	330,295,309,281
173	362,326,859,895	362,112,259,825
174	397,125,074,750	396,895,058,709
175	435,157,697,830	434,911,413,024
176	476,715,857,290	476,452,200,458
177	522,115,831,195	521,833,869,016
178	571,701,605,655	571,400,136,521
179	625,846,753,120	625,524,763,617
180	684,957,390,936	684,613,579,960
181	749,474,411,781	749,107,689,772
182	819,876,908,323	819,485,877,701
183	896,684,817,527	896,268,317,743
184	980,462,880,430	980,019,425,239
185	1,071,823,774,337	1,071,352,143,426
186	1,171,432,692,373	1,170,931,321,289
187	1,280,011,042,268	1,279,478,668,917
188	1,398,341,745,571	1,397,776,745,817
189	1,527,273,599,625	1,526,674,692,637
190	1,667,727,404,093	1,667,092,933,883
191	1,820,701,100,652	1,820,029,801,774
192	1,987,276,856,363	1,986,567,076,583
193	2,168,627,105,469	2,167,877,641,991
194	2,366,022,741,845	2,365,231,996,716
195	2,580,840,212,973	2,580,007,105,134
196	2,814,570,987,591	2,813,694,041,537
197	3,068,829,878,530	3,067,908,214,547
198	3,345,365,983,698	3,344,398,333,926
199	3,646,072,432,125	3,645,058,221,777
200	3,972,999,029,388	3,971,937,310,627
201	4,328,363,658,647	4,327,254,289,804
202	4,714,566,886,083	4,713,409,383,776
203	5,134,205,287,973	5,133,000,108,179
204	5,590,088,317,495	5,588,835,619,535
205	6,085,253,859,260	6,083,954,904,602
206	6,622,987,708,040	6,621,643,524,989
207	7,206,841,706,490	7,205,454,616,281
208	7,840,656,226,137	7,839,228,409,394
209	8,528,581,302,375	8,527,116,458,389
210	9,275,102,575,355	9,273,604,382,976
211	10,085,065,885,767	10,083,539,816,166
212	10,963,707,205,259	10,962,158,865,096
213	11,916,681,236,278	11,915,118,346,825
214	12,950,095,925,895	12,948,526,550,206
215	14,070,545,699,287	14,068,980,396,734
216	15,285,151,248,481	15,283,601,178,668
217	16,601,598,107,914	16,600,077,386,512
218	18,028,182,516,671	18,026,706,182,776
219	19,573,856,161,145	19,572,442,752,352
220	21,248,279,009,367	21,246,948,396,225
221	23,061,871,173,849	23,060,647,367,795
222	25,025,873,760,111	25,024,782,607,963
223	27,152,408,925,615	27,151,481,184,166
224	29,454,549,941,750	29,453,818,819,391
225	31,946,390,696,157	31,945,895,218,868
226	34,643,126,322,519	34,642,908,729,066
227	37,561,133,582,570	37,561,243,002,581
228	40,718,063,627,362	40,718,553,325,339
229	44,132,934,884,255	44,133,866,294,224
230	47,826,239,745,920	47,827,679,552,474
231	51,820,051,838,712	51,822,076,403,342
232	56,138,148,670,947	56,140,840,962,281
233	60,806,135,438,329	60,809,589,881,049
234	65,851,585,970,275	65,855,905,223,838
235	71,304,185,514,919	71,309,485,804,774
236	77,195,892,663,512	77,202,300,401,270
237	83,561,103,925,871	83,568,761,572,559
238	90,436,839,668,817	90,445,902,217,271
239	97,862,933,703,585	97,873,575,120,469
240	105,882,246,722,733	105,894,656,301,136
241	114,540,884,553,038	114,555,274,014,365
242	123,888,443,077,259	123,905,042,795,369
243	133,978,259,344,888	133,997,326,179,546
244	144,867,692,496,445	144,889,505,925,148
245	156,618,412,527,946	156,643,283,280,695
246	169,296,722,391,554	169,324,988,512,763
247	182,973,889,854,026	183,005,926,250,793
248	197,726,516,681,672	197,762,731,133,712
249	213,636,919,820,625	213,677,763,545,749
250	230,793,554,364,681	230,839,518,036,227
251	249,291,451,168,559	249,343,076,603,055
252	269,232,701,252,579	269,290,577,482,513
253	290,726,957,916,112	290,791,734,156,514
254	313,891,991,306,665	313,964,373,128,410
255	338,854,264,248,680	338,935,027,975,330
256	365,749,566,870,782	365,839,555,950,559
257	394,723,676,655,357	394,823,817,645,196
258	425,933,084,409,356	426,044,383,633,593
259	459,545,750,448,675	459,669,311,792,618
260	495,741,934,760,846	495,878,956,707,614
261	534,715,062,908,609	534,866,858,363,775
262	576,672,674,947,168	576,840,668,810,756
263	621,837,416,509,615	622,023,167,767,158
264	670,448,123,060,170	670,653,323,043,735
265	722,760,953,690,372	722,987,450,753,395
266	779,050,629,562,167	779,300,428,193,871
267	839,611,730,366,814	839,887,018,776,921
268	904,760,108,316,360	905,063,258,652,871
269	974,834,369,944,625	975,167,969,139,993
270	1,050,197,489,931,110	1,050,564,341,278,630
271	1,131,238,503,938,600	1,131,641,661,657,750
272	1,218,374,349,844,330	1,218,817,122,296,590
273	1,312,051,800,816,210	1,312,537,789,269,910
274	1,412,749,565,173,450	1,413,282,669,049,830
275	1,520,980,492,851,170	1,521,564,953,213,890
276	1,637,293,969,337,170	1,637,934,376,582,770
277	1,762,278,433,057,260	1,762,979,775,792,390
278	1,896,564,103,591,580	1,897,331,779,227,310
279	2,040,825,852,575,070	2,041,665,722,301,010
280	2,195,786,311,682,510	2,196,704,714,572,330
281	2,362,219,145,337,710	2,363,222,960,201,970
282	2,540,952,590,045,690	2,542,049,253,703,810
283	2,732,873,183,547,530	2,734,070,760,525,010
284	2,938,929,793,929,550	2,940,236,999,631,600
285	3,160,137,867,148,990	3,161,564,146,448,330
286	3,397,584,011,986,770	3,399,139,568,230,940
287	3,652,430,836,071,050	3,654,126,719,723,870
288	3,925,922,161,489,420	3,927,770,305,994,760
289	4,219,388,528,587,090	4,221,401,850,472,780
290	4,534,253,126,900,880	4,536,445,569,650,230
291	4,872,038,056,472,080	4,874,424,703,634,060
292	5,234,371,069,753,670	5,236,968,198,243,810
293	5,622,992,691,950,600	5,625,817,899,891,690
294	6,039,763,882,095,510	6,042,836,153,121,440
295	6,486,674,127,079,080	6,490,013,974,965,660
296	6,965,850,144,195,830	6,969,479,689,946,320
297	7,479,565,078,510,580	7,483,508,214,051,230
298	8,030,248,384,943,040	8,034,530,865,140,210
299	8,620,496,275,465,020	8,625,145,903,435,480
300	9,253,082,936,723,600	9,258,129,673,182,110
301	9,930,972,392,403,500	9,936,448,565,599,450
302	10,657,331,232,548,800	10,663,271,667,656,800
303	11,435,542,077,822,100	11,441,984,334,851,100
304	12,269,218,019,229,400	12,276,202,545,689,500
305	13,162,217,895,057,700	13,169,788,295,603,300
306	14,118,662,665,280,000	14,126,865,881,272,200
307	15,142,952,738,857,100	15,151,839,354,143,200
308	16,239,786,535,829,600	16,249,410,987,302,400
309	17,414,180,133,147,200	17,424,601,057,574,000
310	18,671,488,299,600,300	18,682,768,780,005,900
311	20,017,426,762,576,900	20,029,634,721,655,400
312	21,458,096,037,352,800	21,471,304,525,099,900
313	23,000,006,655,487,300	23,014,294,295,620,400
314	24,650,106,150,830,400	24,665,557,475,818,700
315	26,415,807,633,566,300	26,432,513,591,256,500
316	28,305,020,340,996,000	28,323,078,684,208,600
317	30,326,181,989,842,900	30,345,697,851,320,000
318	32,488,293,351,466,600	32,509,379,696,109,400
319	34,800,954,869,440,800	34,823,733,146,946,400
320	37,274,405,776,748,000	37,299,006,445,639,900
321	39,919,565,526,999,900	39,946,128,795,469,600
322	42,748,078,035,954,600	42,776,754,468,280,500
323	45,772,358,543,578,000	45,803,309,901,036,700
324	49,005,643,635,237,800	49,039,043,576,859,200
325	52,462,044,228,828,600	52,498,079,266,012,700
326	56,156,602,112,874,200	56,195,472,418,011,800
327	60,105,349,839,666,500	60,147,270,329,785,500
328	64,325,374,609,114,500	64,370,575,877,957,000
329	68,834,885,946,073,800	68,883,615,494,065,300
330	73,653,287,861,850,300	73,705,811,169,211,700
331	78,801,255,302,666,600	78,857,857,225,508,800
332	84,300,815,636,225,100	84,361,801,640,598,500
333	90,175,434,980,549,600	90,241,132,726,932,600
334	96,450,110,192,202,700	96,520,870,953,318,000
335	103,151,466,321,735,000	103,227,666,780,563,000
336	110,307,860,425,292,000	110,389,904,302,398,000
337	117,949,491,546,113,000	118,037,811,639,875,000
338	126,108,517,833,796,000	126,203,577,886,370,000
339	134,819,180,623,301,000	134,921,477,635,335,000
340	144,117,936,527,873,000	144,228,002,896,388,000
341	154,043,597,379,576,000	154,162,003,523,593,000
342	164,637,479,165,761,000	164,764,835,973,643,000
343	175,943,559,810,422,000	176,080,521,617,844,000
344	188,008,647,052,292,000	188,155,914,441,466,000
345	200,882,556,287,683,000	201,040,879,464,350,000
346	214,618,299,743,286,000	214,788,481,736,150,000
347	229,272,286,871,217,000	229,455,187,360,548,000
348	244,904,537,455,382,000	245,101,076,427,177,000
349	261,578,907,351,144,000	261,790,069,437,081,000
350	279,363,328,483,702,000	279,590,167,131,561,000
351	298,330,063,062,758,000	298,573,705,454,962,000
352	318,555,973,788,329,000	318,817,625,598,558,000
353	340,122,810,048,577,000	340,403,761,014,690,000
354	363,117,512,048,110,000	363,419,141,393,747,000
355	387,632,532,919,029,000	387,956,315,666,778,000
356	413,766,180,933,342,000	414,113,694,080,277,000
357	441,622,981,929,358,000	441,995,911,597,065,000
358	471,314,064,268,398,000	471,714,212,733,254,000
359	502,957,566,506,000,000	503,386,860,294,917,000
360	536,679,070,310,691,000	537,139,568,199,682,000
361	572,612,058,898,037,000	573,105,961,076,774,000
362	610,898,403,751,884,000	611,428,060,918,555,000
363	651,688,879,997,206,000	652,256,803,730,406,000
364	695,143,713,458,946,000	695,752,586,553,637,000
365	741,433,159,884,081,000	742,085,848,086,667,000
366	790,738,119,649,411,000	791,437,683,397,883,000
367	843,250,788,562,528,000	844,000,496,260,715,000
368	899,175,348,396,088,000	899,978,689,741,625,000
369	958,728,697,912,338,000	959,589,398,908,019,000
370	1,022,141,228,367,340,000	1,023,063,266,444,290,000
371	1,089,657,644,424,390,000	1,090,645,265,413,150,000
372	1,161,537,834,849,960,000	1,162,595,570,132,140,000
373	1,238,057,794,119,120,000	1,239,190,479,809,280,000
374	1,319,510,599,727,470,000	1,320,723,396,115,790,000
375	1,406,207,446,561,480,000	1,407,505,859,788,460,000
376	1,498,478,743,590,580,000	1,499,868,647,676,860,000
377	1,596,675,274,490,750,000	1,598,162,935,821,660,000
378	1,701,169,427,975,810,000	1,702,761,530,251,060,000
379	1,812,356,499,739,470,000	1,814,060,171,624,690,000
380	1,930,656,072,350,460,000	1,932,478,915,720,990,000
381	2,056,513,475,336,630,000	2,058,463,596,496,840,000
382	2,190,401,332,423,760,000	2,192,487,374,066,330,000
383	2,332,821,198,543,890,000	2,335,052,374,987,350,000
384	2,484,305,294,265,410,000	2,486,691,427,602,110,000
385	2,645,418,340,688,760,000	2,647,969,900,547,130,000
386	2,816,759,503,217,940,000	2,819,487,647,630,880,000
387	2,998,964,447,736,450,000	3,001,881,067,996,670,000
388	3,192,707,518,433,530,000	3,195,825,285,280,280,000
389	3,398,704,041,358,160,000	3,402,036,455,563,610,000
390	3,617,712,763,867,600,000	3,621,274,208,413,110,000
391	3,850,538,434,667,420,000	3,854,344,231,777,640,000
392	4,098,034,535,626,590,000	4,102,101,005,688,210,000
393	4,361,106,170,762,280,000	4,365,450,696,609,950,000
394	4,640,713,124,699,620,000	4,645,354,218,123,350,000
395	4,937,873,096,788,190,000	4,942,830,470,972,180,000
396	5,253,665,124,416,970,000	5,258,959,768,984,820,000
397	5,589,233,202,595,400,000	5,594,887,465,213,220,000
398	5,945,790,114,707,870,000	5,951,827,785,731,630,000
399	6,324,621,482,504,290,000	6,331,067,886,881,070,000
400	6,727,090,051,741,040,000	6,733,972,144,452,060,000
401	7,154,640,222,653,940,000	7,161,986,692,184,490,000
402	7,608,802,843,339,870,000	7,616,644,219,256,550,000
403	8,091,200,276,484,460,000	8,099,569,045,897,860,000

I'm sure the estimate would become more accurate if I threw in more decimal places.

Quote: Wizard

I found a very good estimate for the P(n) if you know P(n-1).

Here it is

P(n) = P(n-1) * exp(0.923266 * n^-0.448858)

{snip}

I'm sure the estimate would become more accurate if I threw in more decimal places.

I think you could come up with a more accurate version of this formula if you derived two different formulas: one each for even and odd values of n. If you use one formula, you should always tend to overestimate for odd n and underestimate for even n.

Quote: gordonm888

I think you could come up with a more accurate version of this formula if you derived two different formulas: one each for even and odd values of n. If you use one formula, you should always tend to overestimate for odd n and underestimate for even n.

Doesn't this odd and even effect diminish over time? I did some calculus to get at a direct formula. However, there is already a good formula for this. I'll put it in spoiler tags. I'll just say that it goes back to what I wrote earlier, that series seems to always come down to pi, e, and primes. I'll add to that the Fibonacci series.

Quote: Wizard

Doesn't this odd and even effect diminish over time? I did some calculus to get at a direct formula. However, there is already a good formula for this. I'll put it in spoiler tags. I'll just say that it goes back to what I wrote earlier, that series seems to always come down to pi, e, and primes. I'll add to that the Fibonacci series.

Partitions estimate

P(n) = 1/(4n*sqrt(3)) * exp(pi * sqrt(2n/3))

I've looked at this only up from n=1 . . . 51. The odd/even effect definitely appears to get larger on an absolute basis but smaller on a % basis. At n=50,51 the P(n) for odd and even ns may be off by +/- 75, but the P(n) is >200,000 so as a % it is a small effect.