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17 members have voted

Wizard
Joined: Oct 14, 2009
• Posts: 21923
August 16th, 2019 at 9:37:15 AM permalink
Quote: gordonm888

What are the results for a divisor of 13?

Damn Excel doesn't handle big numbers very well, so my sample size is rather small:

Divisor Yes No Ratio Expected
2 44 57 43.6% 50.0%
3 37 66 35.9% 33.3%
5 35 72 32.7% 20.0%
7 33 76 30.3% 14.3%
11 52 60 46.4% 9.1%
13 4 109 3.5% 7.7%

So 13's are underrepresented. I think it is indeed significant how often 11 pops up and how often 13 does not.

I think I'll add more code to check the factorization for larger numbers. I can handle numbers up to 2^64 in C++. Projects like this make me jealous of the people who have Mathematica.

Quote:

I have taken all the primes from 2-1,000,000 and converted them into different bases/radices from 2-100+ and looked for frequency of integers in the digits (ignoring the first and last digit of each prime.) It is extraordinarily regular, with a digit frequency that has very little variance around the expected number. Given that as a background, I am surprised to see your large discrepancy in the frequency of divisor 11. It seems like it might be significant.

Can you show me a table to illustrate what you mean?
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Joined: Oct 14, 2009
• Posts: 21923
Thanks for this post from:
August 16th, 2019 at 4:18:57 PM permalink
I took the analysis of how often total partitions are divisible by a prime number up to an initial quantity of 405. Here are the results.

Prime Count in 2 to 405 Ratio Expected
2 181 44.8% 50.0%
3 139 34.4% 33.3%
5 139 34.4% 20.0%
7 101 25.0% 14.3%
11 108 26.7% 9.1%
13 25 6.2% 7.7%
17 17 4.2% 5.9%
23 18 4.5% 4.3%
29 11 2.7% 3.4%

This shows a huge surplus of total partitions divisible by 5, 7, and 11, especially 11. You may recall these are the same primes in the Ramanujan identities. Very interesting...
It's not whether you win or lose; it's whether or not you had a good bet.
kubikulann
Joined: Jun 28, 2011
• Posts: 905
August 17th, 2019 at 1:53:07 PM permalink
« Primey »

- Semi-prime
- k-almost prime
- Sphenic number

https://en.wikipedia.org/wiki/Table_of_prime_factors?wprov=sfti1
Reperiet qui quaesiverit
rsactuary
Joined: Sep 6, 2014
• Posts: 1680
August 17th, 2019 at 3:43:43 PM permalink
A favor of the Wiz:

Can you run your code for partitions of 10, 11, 12, 13, 14 and 15 and PM or email the lists to me please? I don't have the software to run the code myself.
Last edited by: rsactuary on Aug 17, 2019
gordonm888
Joined: Feb 18, 2015
• Posts: 2563
August 18th, 2019 at 9:21:24 PM permalink
Quote: Wizard

Can you show me a table to illustrate what you mean?

Sorry, I spent some time looking through my many spreadsheets and I didn't find the work that I had mentioned. I did find some spreadsheets with the primes from 3-19,997 converted into prime number radices (bases) so I used them to produce the (modest) results shown in the table below.

Explanation: This table is the frequency of digits (as shown in the first column) in prime numbers converted into base 2, 5, 7,and 11 - in which I only count the digits as they appear in the "2nd to last digit" and "3rd to last digit" of each prime number.

So in the 2nd row I am counting the frequency of digits 1-4 as they appear in all the primes from 29-19997, when those primes are written in Base 5 -and only counting the digits in the 5^3 and 5^2 columns of the prime numbers.

As an example the prime number 19,917 is written in base 5 as 1114202, so I count one "2" and one "0" as shown in bold in the number.
Digits
Base 3: 11-19997
Base 5: 29-19997
Base 7: 53-19997
Base 11: 127-19997
0
1492
885
661
410
1
1503
917
627
409
2
1517
916
633
394
3
898
634
412
4
885
649
396
5
641
404
6
645
414
7
417
8
423
9
388
10
393

This was just a quick effort to show the kind of work that I had done. The original work was on every prime less than 1,000,000 and the variance in the calculated frequencies of various digits seemed pretty damn small, as I remember it.
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
Wizard
Joined: Oct 14, 2009
• Posts: 21923
August 19th, 2019 at 5:04:59 AM permalink
Quote: rsactuary

A favor of the Wiz:

Can you run your code for partitions of 10, 11, 12, 13, 14 and 15 and PM or email the lists to me please? I don't have the software to run the code myself.

n Partitions of n Factorization of Partitions of n
222
333
455
577
61111
7153,5
8222,11
9302,3,5
10422,3,7
11562,2,2,7
12777,11
13101101
141353,3,3,5
151762,2,2,2,11
162313,7,11
172973,3,3,11
183855,7,11
194902,5,7,7
206273,11,19
217922,2,2,3,3,11
2210022,3,167
2312555,251
2415753,3,5,5,7
2519582,11,89
2624362,2,3,7,29
2730102,5,7,43
2837182,11,13,13
2945655,11,83
3056042,2,3,467
3168422,11,311
3283493,11,11,23
33101433,3,7,7,23
34123102,5,1231
35148833,11,11,41
361797717977
37216377,11,281
38260155,11,11,43
39311853,3,3,3,5,7,11
40373382,3,7,7,127
41445833,7,11,193
42531742,11,2417
43632613,3,3,3,11,71
44751755,5,31,97
45891342,41,1087
461055582,3,73,241
471247542,7,7,19,67
481472733,7,7013
491735255,5,11,631
502042262,11,9283
512399433,11,11,661
522815893,7,11,23,53
533299313,3,7,5237
543861555,7,11,17,59
554512762,2,7,71,227
5652682311,47,1019
576141542,3,102359
587152202,2,5,11,3251
598318202,2,5,11,19,199
6096646717,139,409
6111215053,5,7,11,971
6213001562,2,11,13,2273
6315054993,113,4441
6417416302,5,11,71,223
6520125582,1006279
6623235202,2,2,2,2,2,5,53,137
6726796891181,2269
6830877353,5,7,7,4201
6935543455,641,1109
7040879682,2,2,2,2,3,97,439
7146972053,5,313147
72539278311,139,3527
73618568923,131,2053
7470895002,2,5,5,5,11,1289
7581182642,2,2,3,7,11,23,191
7692890917,1327013
771061986310619863
78121321642,2,11,103,2677
79138486502,5,5,173,1601
80157964762,2,3,3,227,1933
811800432711,1636757
82205062555,7,7,7,11,1087
83233384697,11,303097
84265436602,2,5,11,13,9281
853016735711,11,249317
86342629622,23,37,41,491
873888767311,3535243
88441081093,3,83,137,431
89499959255,5,7,7,40813
90566341732473,22901
916411235929,373,5927
92725338073371,21517
938201017759,1390003
94926697202,2,2,5,11,13,17,953
95104651419283,369793
961181143042,2,2,2,2,2,2,2,2,2,2,7,7,11,107
971332309302,3,5,7,29,131,167
981501981362,2,2,11,1706797
991692298755,5,5,1353839
1001905692922,2,43,59,89,211
1012144811262,31,3459373
1022412653793,2423,33191
1032712489502,5,5,7,774997
1043048013653,5,11,1847281
10534232570911,43,43,16831
1063842763362,2,2,2,3,8005757
1074311493893,11,173,75521
1084835028442,2,11,10988701
1095419462402,2,2,2,2,2,2,5,11,23,3347
1106071637462,7,4049,10711
1116799032033,7,67,483229
1127610021562,2,190250539
1138513766282,2,212844157
1149520506655,193,986581
11510641444513,61,67,229,379
11611889082482,2,2,11,11,157,7823
11713277100762,2,7,7,11,615821
118148207414311,197,827,827
11916536686655,11,30066703
12018443495602,2,2,5,47,981037
1212056148051461,4460191
12222913209122,2,2,2,9013,15889
123255233824179,32308079
12428419405002,2,3,5,5,5,7,31,8731
12531631273522,2,2,3,7,11,59,67,433
12635192226922,2,89,379,26083
12739138642955,11,67,1062107
12843510786002,2,2,5,5,11,17,317,367
12948352718702,3,3,5,11,13,157,2393
13053713154002,2,2,5,5,11,157,15551
13159645395042,2,2,2,7,7,7,11,29,3407
13266208308896620830889
13373466295122,2,2,3,3,1319,77359
13481490406955,17,89,1077203
13590358360762,2,59,569,67289
136100155816802,2,2,2,5,13,31,41,7577
137110976450162,2,2,17,1367,59693
138122923418313,3,7,7,11,733,3457
139136109498955,79,34458101
140150658781355,7,11,39132151
141166706892082,2,2,29,31,991,2339
142184402933202,2,2,3,3,5,127,403331
143203909827577589,2686913
144225406544455,5807,776327
145249088580097,7,53,73,83,1583
1462751705259953197,517267
147303886719782,3,7,24151,29959
1483354941949711,73,41780099
149370273552002,2,2,2,2,2,5,5,11,11,11,17387
1504085323531311,17,197,1108967
151450606245822,3,7510104097
152496862884217,11,751,859223
153547703363242,2,11,34513,36067
154603566732802,2,2,2,2,2,2,2,5,37,1274423
1556649318209719,8087,432749
15673232243759463,1777,89009
1578063096476980630964769
158887517788022,79,691,853,953
159976627285555,7,29,67,1436111
1601074381594662,3,3,11,443,1224869
161118159068427797,148254791
1621299139046373,3,11,127,10332769
1631427989959302,3,5,4759966531
1641569194752953,3,5,11,14867,21323
1651723898002555,313,1543,71389
1661893348225797,7,37,53,1277,1543
1672078904201022,19,73,503,148991
168228204732751228204732751
1692504389251153,3,3,5,1855103149
1702747686171302,5,7,47,83516297
1713013848020482,2,2,2,2,2,2,2,11,11,1609,6047
172330495499613103,2351,1364821
1733623268598953,5,7,11,13,37,652189
1743971250747502,5,5,5,103,1627,9479
1754351576978302,5,43515769783
1764767158572902,5,443,107610803
1775221158311955,7,97,153789641
1785717016056553,5,17,2241967081
1796258467531202,2,2,2,2,5,277,3467,4073
1806849573909362,2,2,3,7,7,13,59,643,1181
1817494744117813,249824803927
18281987690832311,23,27967,115873
18389668481752761,293,50169799
1849804628804302,5,11,107,83301859
18510718237743373469,6653,46441
18611714326923731171432692373
18712800110422682,2,7,7,18713,348991
18813983417455711398341745571
18915272735996253,3,3,3,5,5,5,61,127,19471
1901667727404093317,7283,722363
19118207011006522,2,3,7,11171,1940293
192198727685636323,503,171776027
19321686271054693,11,71,269,3440807
19423660227418455,7,7,521,18535961
19525808402129733,11,78207279181
19628145709875913,11,89,958314943
19730688298785302,5,13,43,257,2136131
19833453659836982,3,3,185853665761
19936460724321253,3,5,5,5,3240953273
20039729990293882,2,3,331083252449
20143283636586477,19,23,47,4441,6779
20247145668860833,837673,1876057
2035134205287973151,34001359523
20455900883174955,11,19,5349366811
20560852538592602,2,5,304262692963
20666229877080402,2,2,5,11,15052244791
20772068417064902,3,3,5,829,5399,17891
20878406562261373,7,373364582197
20985285813023753,5,5,5,19,197,2083,2917
21092751025753555,487,1091,3491363
211100850658857673,17,61,227,14280811
2121096370720525910963707205259
213119166812362782,7,13,31,79,131,409,499
214129500959258953,5,2351,7573,48491
215140705456992877,7,11,11,23497,100999
2161528515124848115285151248481
217166015981079142,2140069,3878753
2181802818251667153,340154387107
219195738561611455,11,13,313,87463331
220212482790093673,7082759669789
221230618711738493,223,138283,249287
222250258737601113,7,17,1259,55679497
223271524089256153,5,23,353,4801,46439
224294545499417502,5,5,5,16301,7227667
225319463906961573,7,37,97,2887,146819
226346431263225193,11,13,13,79,78630193
227375611335825702,5,13,173,449,3719657
228407180636273622,11,23,80470481477
229441329348842555,7,7,191,943112189
230478262397459202,2,2,2,2,2,2,5,109843,680321
231518200518387122,2,2,3,59,59,620272573
232561381486709473,7,32009,83515423
23360806135438329307,347,9749,58549
234658515859702755,5,7,376294776973
235713041855149197,10186312216417
236771958926635122,2,2,7,7,263,748776797
2378356110392587111,11,13,17,677,4615703
238904368396688173,7591,3971230829
239978629337035853,5,1114697,5852887
2401058822467227333,13,13,83,2516153293
2411145408845530382,57270442276519
2421238884430772597,31,570914484227
2431339782593448882,2,2,7,7,97,5783,609289
2441448676924964455,14813,1955953453
2451566184125279462,4007,19543101139
2461692967223915542,431,18433,10654799
2471829738898540262,77899,1174430287
2481977265166816722,2,2,11,79,28441673861
2492136369198206253,3,3,5,5,5,5,7,277,6529121
2502307935543646813,7,11,10037,99542323
251249291451168559887,281050114057
2522692327012525794177,64455997427
2532907269579161122,2,2,2,53,73,4696416353
2543138919913066655,17,643,9649,595207
2553388542642486802,2,2,5,631,859,1429,10937
2563657495668707822,91381,2001234211
2573947236766553573,7,18796365555017
2584259330844093562,2,545863,195073253
2594595457504486753,5,5,11,234197,2378447
2604957419347608462,31,53,150864861461
26153471506290860911,41,349,20389,166619
2625766726749471682,2,2,2,2,19,439,2160534839
2636218374165096155,419,29867,9938051
2646704481230601702,5,7,7,13,41,1907,1346143
2657227609536903722,2,419,8167,52802941
2667790506295621673,3,3,28853727020821
2678396117303668142,3,53,6857,385050089
2689047601083163602,2,2,3,3,5,2513222523101
2699748343699446255,5,5,173,45079046009
27010501974899311173,11,83,47309,8104667
27111312385039386062,7,3041,26571111569
272121837434984433311,317333,349038091
27313120518008162155,499,2003,9829,26711
27414127495651734502,5,5,7,7,173,4327,770311
27515209804928511755,5,60839219714047
2761637293969337171167,1777,1931,2857199
27717622784330572693,3,29,6752024647729
27818965641035915842,2,2,2,2,3,7,7,7,23,2504230711
27920408258525750755,5,373,218855319311
28021957863116825162,2,7,389,3853,52321891
28123622191453377113,11,37,983809,1966499
28225409525900456982,29,59,619,1199570861
28327328731835475355,7,7,53,210463857031
28429389297939295553,5,790189,247951633
28531601378671489977,7,7,52517,175433287
28633975840119867733,3,3,7,43,547,764280217
287365243083607105373,50033299124261
28839259221614894222,3,103,2162249,2937971
28942193885285870953,5,263,1069553492671
29045342531269008862,3,227,9349,356093047
29148720380564720842,2,1218009514118021
29252343710697536722,2,2,7,7,11,97,223,56118901
29356229926919506055,7087043,158683747
29460397638820955155,11,13,73,1187,7027,13873
29564866741270790882,2,2,2,461,1033,851335711
29669658501441958313,7,83,5783,691072399
29774795650785105842,2,2,3,3,3413,6971,4366289
29880302483849430402,2,2,2,2,2,3,5,4801,1742312449
29986204962754650255,5,7,17,2897645806879
30092530829367236022,137,1021,33075784213
30199309723924035013,6491,509986771037
3021065733123254883910657331232548839
303114355420778221042,2,2,11,53,2451874373461
304122692180192294653,5,7,257,617,1249,589993
305131622178950577042,2,2,11,117101,1277279083
306141186626652800055,7,23,67,79,2357,1405841
307151429527388571942,3,3,41,607,33803799259
3081623978653582966311149,68473,21272819
309174141801331472955,12650773,275306183
310186714882996003642,2,181,503,51271070537
311200174267625769455,13,31,199,33479,1491103
312214580960373528917,15493,217027,911683
313230000066554873377,7,7,41,1635497877799
314246501061508304902,3,3,5,11,59,422018595289
315264158076335663262,691,1609,11879544977
316283050203409960033,3,11,113,233,10859101993
317303261819898429642,2,557,44111,308571383
318324882933514666542,13,3881,321965922259
319348009548694408302,3,5,7,165718832711623
320372744057767480777,109,48852432210679
321399195655269999917,5702795075285713
322427480780359546962,2,2,3,37,48139727517967
323457723585435780282,2,47279,415343,582731
324490056436352378755,5,5,17,19,1213762071461
3255246204422882864111,79,1013,38749,1537997
3265615660211287428913,701,6162251960153
327601053498396665442,2,2,2,7,7,17,79,277,283,728207
328643253746091145502,5,5,73,17623390303867
329688348859460738502,5,5,523,661673,3978263
330736532878618503393,24551095953950113
331788012553026666155,463,701,1033,1069,43973
3328430081563622511929,31,151,37447,16583573
3339017543498054962329,67,149,2819,110492831
334964501101922027602,2,2,5,7,7,43,61,86981,215687
3351031514663217353255,5,13,12161,26098933241
3361103078604252927722,2,3,11,41,341743,59641567
3371179494915461139722,2,20533,1436096668121
3381261085178337963555,11,13,176375549417897
3391348191806233015202,2,2,2,3,5,561746585930423
3401441179365278738322,2,2,3,3,3,109,6121217147803
3411540435973795760302,3,5,7,7,1079531,97071379
3421646374791657610442,2,3,3,19,8761,27473812231
34317594355981042275323,76253,178897,560771
3441880086470522929802,2,5,10937,617657,1391561
34520088255628768315979,113,1156801,19452617
3462146182997432862997,5923,28069,184416611
3472292722868712171502,3,3,3,5,5,11,2311,9257,721697
3482449045374553824062,3,7,17,23,14913197993873
3492615789073511441255,5,5,23,523,173965521557
3502793633284837021522,2,2,3,47,349,997,711770053
3512983300630627580762,2,3,7,106783,33259492333
3523185559737883290842,2,3391,382231,61442951
3533401228100485774282,2,3,239,118592332652921
3543631175120481100053,5,29,47,67,971,273002137
3553876325329190292237,24113,2296523706353
3564137661809333423622,101,3557,15199,37888267
3574416229819293584373,13,13,871051246408991
3584713140642683987802,2,3,3,3,5,11,11,13,401,1383705109
3595029575665060000202,2,5,101,349,104459,6829811
36053667907031069112111,14423,3382722484357
3615726120588980375593,3,131,485676046563221
3626108984037518841017,7,4793,2601150503293
3636516888799972069593,433,4793,104670363037
3646951437134589460402,2,2,5,43,4171819,96877003
3657414331598840816842,2,3,2137,89491,323077621
366790738119649411319790738119649411319
367843250788562528427599,1237,1138047093529
36889917534839608834973,101,179,382429,1781543
3699587286979123380453,5,7,11,751,1105283803889
37010221412283673453622,49556849,10312814969
37110896576444243997822,71,7673645383270421
37211615378348499628502,3,3,5,5,7,547,757,890511641
37312380577941191250853,5,683,722639,167227447
37413195105997274735002,2,3,5,5,5,983,1831,5101,95813
37514062074465614840542,3347,210069830678441
37614984787435905810817,7,431,1949,66863,544477
37715966752744907567912389,16197169,41263051
37817011694279758135255,5,197,331031,1043452463
37918123564997394729502,5,5,73,127,5081,769480909
38019306560723504658122,2,11,43878547098874223
38120565134753366338053,5,227,2801473,215589697
38221904013324237651313,11,66375797952235307
38323328211985438923362,2,2,2,7,7,53,227,177043,1396963
38424843052942654181802,2,5,31,997,4019001026087
385264541834068876370143,11738107,5241165101
38628167595032179427922,2,2,67,2333,8963,12421,20233
38729989644477364521942,131,1607,14149,14449,34841
38831927075184335328262,1596353759216766413
38933987040413581602755,5,13,43,20921,11624628749
39036177127638676044237,7,25439,2902271029993
39138505384346674291862,11,37717,4640466482039
39240980345356265947913,3,137,167,619,32151809299
39343611061707622841142,367,569,1283,21487,378779
39446407131246996235153,5,17,359,78401,646589267
39549378730967881916555,6553,46451,3244402177
39652536651244169751633,3,49789639,11724137413
39755892332025954044882,2,2,7,23,31,37,3783318894683
39859457901147078745973,3,660643346078652733
39963246214825042943253,3,5,5,191,147169784351467
40067270900517410419262,23869,140916880718527
401715464022265394232171,2381,151273,279774827
40276088028433398792693,11,7151,32243012604043
40380912002764844655817,3911,10093,29282410321
40486035517593486550602,2,5,7,61453941138204679
40591476790688591176022,3,34286363,44467043009
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Joined: Oct 14, 2009
• Posts: 21923
August 19th, 2019 at 5:05:49 AM permalink
Quote: gordonm888

Sorry, I spent some time looking through my many spreadsheets and I didn't find the work that I had mentioned. I did find some spreadsheets with the primes from 3-19,997 converted into prime number radices (bases) so I used them to produce the (modest) results shown in the table below...

Thank you. I can't think of a good comment offhand.
It's not whether you win or lose; it's whether or not you had a good bet.
gordonm888
Joined: Feb 18, 2015
• Posts: 2563
Thanks for this post from:
August 19th, 2019 at 8:30:28 AM permalink
In this and the following posts I briefly describe some of the research I am doing on partitions

Partitions in a linear sequence
It is well known that if you have 5 objects that you can partition them into these seven arrangements:
1-1-1-1-1; 2-1-1-1; 2-2-1; 3-1-1; 3-2; 4-1; 5

Now let’s consider the 13 ranks in a standard deck of cards, and for the moment let’s define an Ace to be a high card only. (Equivalently, we could define an ace as a low card only with identical results)
Given: All the various combinations of 5 cards of different rank, and ignoring the suits of the cards
Defining: a 5 card straight as 5 consecutive ranks, a 4 card straight as 4 consecutive ranks, etc. ranging down to a 1-card straight which is a card that has a rank with both adjacent ranks empty, let’s look at some combinations of 5 different ranks

Ex: QJ962. Graphically, this hand looks like this: __QJ_9__6___2. It is easy to see that it has one 2- card straight and three 1-card straights. An obvious way to label this hand (from a connect-ness or straightness point of view) is 2-1-1-1

Ex: T7654. Graphically, this hand is ____T__7654__. It is one 4-card straight and one 1-card straight, which we label as 4-1.
Clearly, all the “straight patterns” are equivalent to the partitions of 5, because we are literally filling 5 of 13 ‘slots’ and noting how the 5 objects are partitioned.

Again, in the cards analogy I am defining aces as either high or low, but not both. However, we can define a mathematical relationship that is more general than cards:

Define a linear string(or array) of 13 slots such that each of the 11 interior slots is each connected to two adjacent slots and such that the two end slots are only connected to one adjacent slot each. Now consider every possible combination of ways to populate the 13 slots with 5 objects. There will be c(13,5) =1287 different combinations of ways to populate the 13 slots with 5 objects.

Surprisingly, given this definition, we can calculate the number of combinations that correspond to the various partitions of 5, equivalently we say that we can calculate the frequencies or probability densities of the partitions of 5 when in a linear string (without loops) of 13 spaces.
Case
1-1-1-1-1
2-1-1-1
2-2-1
3-1-1
3-2
4-1
5
5 into 13, line
126
504
252
252
72
72
9

Of course, there is nothing fundamental about having a string length of 13. Here are some combination frequencies for partitions of 5 when the length of the string is 8-13 available slots.

Case
1-1-1-1-1
2-1-1-1
2-2-1
3-1-1
3-2
4-1
5
5 into 13, line
126
504
252
252
72
72
9
5 into 12, line
56
280
168
168
56
56
8
5 into 11, line
21
140
105
105
42
42
7
5 into 10, line
6
62
58
60
30
30
6
5 into 9, line
1
20
30
30
20
20
5
5 into 8, line
0
4
12
12
12
12
4

A couple of observations based on the above table:
1. The partitions 4-1 and 3-2 appear to be equally likely when 5 objects are placed in a linear string.
2. The “5 into 8” case shows an uncanny symmetry in the number of combinations for the partitions. I have yet to see anything that matches it in the various cases I have analyzed.

Partitions in a Closed Linear String
Consider the game Clock Solitaire where all 13 ranks are arrayed a circle. Further, define the existence of a “straight” in this circular configuration to include KA2 as a 3-card straight, QKA23 as a 5-card straight, etc. More generally consider a closed linear string such that the ends of the string are adjacent (as in a loop or circle) and such that every grid spot in the string has two connections; i.e. no ends.

Given this configuration, the partition frequencies are different as is shown in the table below.
Case
1-1-1-1-1
2-1-1-1
2-2-1
3-1-1
3-2
4-1
5
5 into 13, open line
141
497
244
252
72
72
9
5 into 13, closed (loop) line
92
455
272
274
91
90
13

The difference in these two cases has been illustrated by geometric differences in the configuration of the linear array, but the essential differences in the case arise not from geometry but from the definitions of adjacency or connect-ness.

Partitions of 3
The number 3 has the following three partitions: 1-1-1, 2-1, and 3. It’s a very simple set of partitions, but a good starting point for becoming familiar with this kind of analysis For game analysts, remembering that Ace is either high only or low only, this is “You are dealt 3 unpaired cards, how often do you have a 3-card straight, a 2-card straight, or no connected ranks at all (1-1-1).”

Case
1-1-1
2-1
3
Total
3 of 13, line
165
110
11
286
3 of 12, line
120
90
10
220
3 of 11, line
84
72
9
165
3 of 10, line
56
56
8
120
3 of 9, line
35
42
7
84
3 of 8, line
20
30
6
56
3 of 7, line
10
20
5
35
3 of 6, line
4
12
4
20

Now, let’s look at the those same results for partitions of three when the line is a closed loop (no end points)

Case
1-1-1
2-1
3
Total
3 of 13, closed loop
157
116
13
286
3 of 12, closed loop
112
97
12
220
3 of 11, closed loop
77
77
11
165
3 of 10, closed loop
50
60
10
120
3 of 9, closed loop
30
45
9
84
3 of 8, closed loop
16
32
8
56
3 of 7, closed loop
7
21
7
35
3 of 6, closed loop
2
12
6
20

There appears to be a lot of symmetry in this closed loop version of partitions of 3. At this point, I have not analyzed closed loop systems very much because I have become intrigued by some aspects of possible applications of the “open line” partition frequencies.

Partitions of 4.
The number 4 has the following five partitions: 1-1-1-1, 2-1-1, 2-2, 3-1, and 4. Just for chuckles, let’s see the frequencies of these partitions for open linear arrays of length 8-13.

Case
1-1-1-1
2-1-1
2-2
3-1
4
Total
4 of 13, line
210
360
45
90
10
715
4 of 12, line
126
252
36
72
9
495
4 of 11, line
70
168
28
56
8
330
4 of 10, line
35
105
21
42
7
210
4 of 9, line
15
60
15
30
6
126
4 of 8, line
5
30
10
20
5
70

I always like to look for patterns in numbers, look for primes and special numbers. But when I first calculated these partition frequencies/combinations they frustrated me. The only prime numbers are trivial. I learned to dig into them deeper, which I’ll discuss later in the next few posts.
Last edited by: gordonm888 on Aug 19, 2019
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
gordonm888
Joined: Feb 18, 2015
• Posts: 2563
Thanks for this post from:
August 19th, 2019 at 8:33:50 AM permalink
Calculating Partition Frequency

I will explain an efficient way of calculating the frequency of any partition of the number m when chosen from an open linear array of n objects.

Let me define an example based on card games. Example: You are designing a bonus payout for a card game in which a player is dealt 7 cards and in which the Ace counts only as a high or low card and in which you are ignoring flushes. You want to evaluate a bonus payout for the player when his 7-card hand has no pairs and has two straights that are exactly 3-cards long. There are c(13,7) = 1716 possible combinations of 7 ranks, but how often will 7 cards of different ranks be partitioned into a 3-3-1 partition?

In order to implement the methodology I will describe, we must define for any given partition a TOTAL, a LENGTH and the NUMBER OF PERMUTATIONS

For 3-3-1, The total is 3+3+1=7. The length is 3, because this partition has 3 elements or clumps. And the number of permutations is 3, because it can be sequenced as followed: 3-3-1; 3-1-3; and 1-3-3.
Of specific importance to this calculation is the partition 3-3-1 has 3 permutations.

The next step is to examine the partition of the spaces between the 3 substrings that make up 3-3-1. I call this “partitioning the void.” When selecting 7 objects from a total of 13, you can define the 7 objects that were selected equally well by defining the 6 objects that you have not selected. When picking 7 card ranks from 13, we can equally well think of this as defining 6 card ranks that are NOT SELECTED. This is the reason that c(13,7) =c(13,6) and more generally, that

c(n,m) = c(n,n-m).

Now, given 7 selected from 13, and given the 7 objects will be partitioned as 3-3-1, we know that there must be 6 objects that were not selected and the 6 unselected objects must have a partition that has a length in the range of 2-4. Clearly, the unselected objects must have at least two ‘clumps ‘ so as to divide the 7 objects into three clumps. Similarly, if the 6 unselected objects were partitioned into 5 or more clumps (2-1-1-1, or 1-1-1-1-1-1) they would have to divide the 7 selected objects into more than 3 clumps.

So given an open linear string of n objects and selecting m objects with a partition of length l, the unselected objects must total (n-m) and be arrayed as a partition of (n-m) with length in the rangel -1 to l+1.

For our example, which involves 7 ranks selected from 13 and arrayed as a partition of 3-3-1, let’s look at the partitions of 6. We’ll order them by their length:

l=6: 1-1-1-1-1-1
l=5: 2-1-1-1-1
l=4: 3-1-1-1; 2-2-1-1
l=3: 4-1-1; 3-2-1; 2-2-2
l=2: 5-1; 4-2; 3-3
l=1: 6

For our example of 7 ranks selected from 13 and arrayed as a partition of 3-3-1, the 6 unselected ranks must be partitioned as either 3-1-1-1; 2-2-1-1, 4-1-1; 3-2-1; 2-2-2, 5-1; 4-2; 3-3 because these are all the possible partitions of 6 that have a length of 2-4.

Now, I define a parameter that will be useful in some applications.
p(n,l) = the sum of the number of permutations for all partitions of n with length l.

Example p(6,3) is the sum of number of permutations for all partitions of 6 with length =3. Referring to the list above we have:
4-1-1 which has 3 permutations
3-2-1 which has 6 permutations
2-2-2 which has 1 permutation

So, p(6,3) = 3+6+1 =10.

Similarly, p(6,6)=1; p(6,5)=5; p(6,4)=10: p(6,2)=5 and p(6,1)=1.

The number of combinations of 7 ranks picked out of 13 that have a partition of 3-3-1 I is the product:

(Number of permutations of 3-3-1) x ( p(6,2) + 2 x p(6,3) + p(6,4) )

(3) x (5 + 2 x 10 +10) = 105.

So, when selecting 7 ranks out of 13 (and restricting the ace to be either high or low) the 7 ranks will be partitioned as 3-3-1 in 105 of 1716 possible combinations.

It is straightforward to calculate values of p(n,l) for small values of n; I have found that it is useful to have precalculated tables of this parameter when working certain kinds of problems.
Last edited by: gordonm888 on Aug 19, 2019
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
gordonm888
Joined: Feb 18, 2015
• Posts: 2563
Thanks for this post from:
August 19th, 2019 at 8:45:47 AM permalink
Permutations and Binary Integers
The binary form of integers is an obvious opportunity to investigate partitions in open linear strings. I have just started to consider this, but here are some preliminary examples and observations.

The decimal integer 137 may be written in binary notation as 10001001. Now notice that 10001001 is a string of 8 digits: 3 ones and 5 zeros. The ones are partitioned as 1-1-1 and the 5 zeros are partitioned as 3-2.

The partitions 1-1-1|3-2 do not uniquely connote 197, there are 2 binary numbers with those partitions. We know it is 2 because 1-1-1 has 1 permutation and 3-2 has 2 permutations and 1x2=2.

The set of binary integers with partitions 1-1-1|3-1 is (145, 197).

As another example, the set of binary integers with partitions 2-1-1|2-1 is (75, 77, 83, 89, 101, 105). Here are some other sets, as denoted by combinations of simple partitions.
2-1|1 = (11, 13)
3-1|1 = (23, 29)
4-1|1 = (47, 61)
5-1|1 = (95, 125)
6-1|1 = (191, 253)
2-1|2 = (19, 25)
3-1|3 = (71, 113)
4-1|3 = (143, 241)
2-1-1|1-1 = (43, 45, 53)

Here are some rules for interpreting the partition nomenclature:
1. The first partition is the partition of the 1’s, the second partition refers to the partition of the 0’s.
2. The number of binary integers that have that specific set of partitions for the 1s and 0s will be equal to the product of the permutations of the two partitions.
3. In order to be a valid binary integer, the length of the zeros partition must be equal to or one less than the length of the ones partition.
4. If the length of the partition of zeros is one less than the length of the partition of the 1s, than all the integers in the set will be ODD. If the length of the two partitions are equal, then all the integers in the set will be EVEN.

So given this symbol: 4-3-1-1|2-2-1 it can be immediately deduced that 36 integers will have that configuration of partitions, and they will all be odd numbers in the range 214 to 215-1.

Now let’s look at partitions when writing binary numbers with some of the leading zeroes. To do this it is necessary to define a number “space” or region. Let’s consider all the integers from 0 to (n13-1) and write them all as having 13 digits.
12 = 0000000001100 which is 2|9-2
8,191 = 11111111111111 which is 13|0
7,621 = 1110111000101 which is 3-3-1-1|3-1-1

Notice that for 7,621 the ones partition has a length l=4, while the zeroes partition has a length that is 3, or l-1. Here again, we see that if the ones partition has a length n then the length of the ones partition is constrained to be n-1, n, or n+1.

Let’s make a definition that in the space 0 to (213-1) that a number is:
SMALL if it is 0 to (212-1)
LARGE if it is 212 to (212-1)

Now if the ones partition of an integer in the space 0 to (213-1) is of length l, then it is easily proven that:
- The integer will be SMALL and ODD if the zeros partition is of length ( l - 1 )
- The integer will be LARGE and EVEN if the zeros partition is of length ( l + 1 )
- The integer will be either SMALL and EVEN or LARGE and ODD if the zeros partition is of length l

Combinations and Factor of Two
Let me write down one particular mathematical identity that came to me as I was working with binary numbers.

m=1 to nΣ c(n,m) = 2n – 1

where, again, c(n,m) is the classic formula for combinations when selecting m from n.

Example: Take n=13
c(13,13) =1
c(13,1) = c(13,12) = 13
c(13,2) = c(13,11) =78
c(13,3) = c(13,10) =286
c(13,4) = c(13,9) =715
c(13,5) = c(13,8) =1287
c(13,6) = c(13,7) =1716

and:
1+ 2 x (13 + 78 + 286 + 715 + 1287 + 1716) = 8,191 = 213 - 1

This simple formula potentially links combination math to other areas of number theory such as Mersenne Primes.

The identity may also be written as:

m=0 to nΣc(n,m) = 2n

Where now the summation starts at m=0.

Interesting, huh?
Last edited by: gordonm888 on Aug 19, 2019
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.