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Forum » Problema săptămânii » Problema standard de partitii
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Autor Mesaj
alex2009
Grup: membru
Mesaje: 288
07 Jan 2011, 03:36

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Problema standard de partitii    [Editează]  [Citează] 

O persoana are o infinitate de monezi de 1 unitate, 10 unitati, 100 unitati si 1000 unitati si doreste sa achizitioneze un produs in valoare de 1997 unitati. In cate moduri poate plati ? Dar pentru un produs de 2011 unitati?


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Student Automatica
gauss
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07 Jan 2011, 03:23

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    [Editează]  [Citează] 

O solutie care mi-a venit prima la mana este cea ce scrie functia generatoare

in care gandim desigur leii ca puteri ale lui x,
seria 1+x+xx+... in loc de 1/(1-x)
si seriile celelalte in loc de factorii ceilalti / fractiile rationale celelalte din definitia lui f
si in sfarsit adunarea leilor revine la adunarea puterilor.
Fiecare partitie a leilor revine la o partitie a puterilor.

Desigur, problema a fost doar reformulata, nu am facut nici un progres in directia solutiei. Dar pusa sub aceasta forma, computerul (si sage) calculeaza repede:

sage: R.<x> = PowerSeriesRing(QQ, default_prec = 2020)
sage: f = 1/(1-x)/(1-x^10)/(1-x^100)/(1-x^1000)
sage: f.coefficients()[1997]
2650
sage: f.coefficients()[2011]
2716
functia generatoare modulo x^2020

sage: f
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + 2*x^15 + 2*x^16 + 2*x^17 + 2*x^18 + 2*x^19 + 3*x^20 + 3*x^21 + 3*x^22 + 3*x^23 + 3*x^24 + 3*x^25 + 3*x^26 + 3*x^27 + 3*x^28 + 3*x^29 + 4*x^30 + 4*x^31 + 4*x^32 + 4*x^33 + 4*x^34 + 4*x^35 + 4*x^36 + 4*x^37 + 4*x^38 + 4*x^39 + 5*x^40 + 5*x^41 + 5*x^42 + 5*x^43 + 5*x^44 + 5*x^45 + 5*x^46 + 5*x^47 + 5*x^48 + 5*x^49 + 6*x^50 + 6*x^51 + 6*x^52 + 6*x^53 + 6*x^54 + 6*x^55 + 6*x^56 + 6*x^57 + 6*x^58 + 6*x^59 + 7*x^60 + 7*x^61 + 7*x^62 + 7*x^63 + 7*x^64 + 7*x^65 + 7*x^66 + 7*x^67 + 7*x^68 + 7*x^69 + 8*x^70 + 8*x^71 + 8*x^72 + 8*x^73 + 8*x^74 + 8*x^75 + 8*x^76 + 8*x^77 + 8*x^78 + 8*x^79 + 9*x^80 + 9*x^81 + 9*x^82 + 9*x^83 + 9*x^84 + 9*x^85 + 9*x^86 + 9*x^87 + 9*x^88 + 9*x^89 + 10*x^90 + 10*x^91 + 10*x^92 + 10*x^93 + 10*x^94 + 10*x^95 + 10*x^96 + 10*x^97 + 10*x^98 + 10*x^99 + 12*x^100 + 12*x^101 + 12*x^102 + 12*x^103 + 12*x^104 + 12*x^105 + 12*x^106 + 12*x^107 + 12*x^108 + 12*x^109 + 14*x^110 + 14*x^111 + 14*x^112 + 14*x^113 + 14*x^114 + 14*x^115 + 14*x^116 + 14*x^117 + 14*x^118 + 14*x^119 + 16*x^120 + 16*x^121 + 16*x^122 + 16*x^123 + 16*x^124 + 16*x^125 + 16*x^126 + 16*x^127 + 16*x^128 + 16*x^129 + 18*x^130 + 18*x^131 + 18*x^132 + 18*x^133 + 18*x^134 + 18*x^135 + 18*x^136 + 18*x^137 + 18*x^138 + 18*x^139 + 20*x^140 + 20*x^141 + 20*x^142 + 20*x^143 + 20*x^144 + 20*x^145 + 20*x^146 + 20*x^147 + 20*x^148 + 20*x^149 + 22*x^150 + 22*x^151 + 22*x^152 + 22*x^153 + 22*x^154 + 22*x^155 + 22*x^156 + 22*x^157 + 22*x^158 + 22*x^159 + 24*x^160 + 24*x^161 + 24*x^162 + 24*x^163 + 24*x^164 + 24*x^165 + 24*x^166 + 24*x^167 + 24*x^168 + 24*x^169 + 26*x^170 + 26*x^171 + 26*x^172 + 26*x^173 + 26*x^174 + 26*x^175 + 26*x^176 + 26*x^177 + 26*x^178 + 26*x^179 + 28*x^180 + 28*x^181 + 28*x^182 + 28*x^183 + 28*x^184 + 28*x^185 + 28*x^186 + 28*x^187 + 28*x^188 + 28*x^189 + 30*x^190 + 30*x^191 + 30*x^192 + 30*x^193 + 30*x^194 + 30*x^195 + 30*x^196 + 30*x^197 + 30*x^198 + 30*x^199 + 33*x^200 + 33*x^201 + 33*x^202 + 33*x^203 + 33*x^204 + 33*x^205 + 33*x^206 + 33*x^207 + 33*x^208 + 33*x^209 + 36*x^210 + 36*x^211 + 36*x^212 + 36*x^213 + 36*x^214 + 36*x^215 + 36*x^216 + 36*x^217 + 36*x^218 + 36*x^219 + 39*x^220 + 39*x^221 + 39*x^222 + 39*x^223 + 39*x^224 + 39*x^225 + 39*x^226 + 39*x^227 + 39*x^228 + 39*x^229 + 42*x^230 + 42*x^231 + 42*x^232 + 42*x^233 + 42*x^234 + 42*x^235 + 42*x^236 + 42*x^237 + 42*x^238 + 42*x^239 + 45*x^240 + 45*x^241 + 45*x^242 + 45*x^243 + 45*x^244 + 45*x^245 + 45*x^246 + 45*x^247 + 45*x^248 + 45*x^249 + 48*x^250 + 48*x^251 + 48*x^252 + 48*x^253 + 48*x^254 + 48*x^255 + 48*x^256 + 48*x^257 + 48*x^258 + 48*x^259 + 51*x^260 + 51*x^261 + 51*x^262 + 51*x^263 + 51*x^264 + 51*x^265 + 51*x^266 + 51*x^267 + 51*x^268 + 51*x^269 + 54*x^270 + 54*x^271 + 54*x^272 + 54*x^273 + 54*x^274 + 54*x^275 + 54*x^276 + 54*x^277 + 54*x^278 + 54*x^279 + 57*x^280 + 57*x^281 + 57*x^282 + 57*x^283 + 57*x^284 + 57*x^285 + 57*x^286 + 57*x^287 + 57*x^288 + 57*x^289 + 60*x^290 + 60*x^291 + 60*x^292 + 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+ 135*x^468 + 135*x^469 + 140*x^470 + 140*x^471 + 140*x^472 + 140*x^473 + 140*x^474 + 140*x^475 + 140*x^476 + 140*x^477 + 140*x^478 + 140*x^479 + 145*x^480 + 145*x^481 + 145*x^482 + 145*x^483 + 145*x^484 + 145*x^485 + 145*x^486 + 145*x^487 + 145*x^488 + 145*x^489 + 150*x^490 + 150*x^491 + 150*x^492 + 150*x^493 + 150*x^494 + 150*x^495 + 150*x^496 + 150*x^497 + 150*x^498 + 150*x^499 + 156*x^500 + 156*x^501 + 156*x^502 + 156*x^503 + 156*x^504 + 156*x^505 + 156*x^506 + 156*x^507 + 156*x^508 + 156*x^509 + 162*x^510 + 162*x^511 + 162*x^512 + 162*x^513 + 162*x^514 + 162*x^515 + 162*x^516 + 162*x^517 + 162*x^518 + 162*x^519 + 168*x^520 + 168*x^521 + 168*x^522 + 168*x^523 + 168*x^524 + 168*x^525 + 168*x^526 + 168*x^527 + 168*x^528 + 168*x^529 + 174*x^530 + 174*x^531 + 174*x^532 + 174*x^533 + 174*x^534 + 174*x^535 + 174*x^536 + 174*x^537 + 174*x^538 + 174*x^539 + 180*x^540 + 180*x^541 + 180*x^542 + 180*x^543 + 180*x^544 + 180*x^545 + 180*x^546 + 180*x^547 + 180*x^548 + 180*x^549 + 186*x^550 + 186*x^551 + 186*x^552 + 186*x^553 + 186*x^554 + 186*x^555 + 186*x^556 + 186*x^557 + 186*x^558 + 186*x^559 + 192*x^560 + 192*x^561 + 192*x^562 + 192*x^563 + 192*x^564 + 192*x^565 + 192*x^566 + 192*x^567 + 192*x^568 + 192*x^569 + 198*x^570 + 198*x^571 + 198*x^572 + 198*x^573 + 198*x^574 + 198*x^575 + 198*x^576 + 198*x^577 + 198*x^578 + 198*x^579 + 204*x^580 + 204*x^581 + 204*x^582 + 204*x^583 + 204*x^584 + 204*x^585 + 204*x^586 + 204*x^587 + 204*x^588 + 204*x^589 + 210*x^590 + 210*x^591 + 210*x^592 + 210*x^593 + 210*x^594 + 210*x^595 + 210*x^596 + 210*x^597 + 210*x^598 + 210*x^599 + 217*x^600 + 217*x^601 + 217*x^602 + 217*x^603 + 217*x^604 + 217*x^605 + 217*x^606 + 217*x^607 + 217*x^608 + 217*x^609 + 224*x^610 + 224*x^611 + 224*x^612 + 224*x^613 + 224*x^614 + 224*x^615 + 224*x^616 + 224*x^617 + 224*x^618 + 224*x^619 + 231*x^620 + 231*x^621 + 231*x^622 + 231*x^623 + 231*x^624 + 231*x^625 + 231*x^626 + 231*x^627 + 231*x^628 + 231*x^629 + 238*x^630 + 238*x^631 + 238*x^632 + 238*x^633 + 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296*x^717 + 296*x^718 + 296*x^719 + 304*x^720 + 304*x^721 + 304*x^722 + 304*x^723 + 304*x^724 + 304*x^725 + 304*x^726 + 304*x^727 + 304*x^728 + 304*x^729 + 312*x^730 + 312*x^731 + 312*x^732 + 312*x^733 + 312*x^734 + 312*x^735 + 312*x^736 + 312*x^737 + 312*x^738 + 312*x^739 + 320*x^740 + 320*x^741 + 320*x^742 + 320*x^743 + 320*x^744 + 320*x^745 + 320*x^746 + 320*x^747 + 320*x^748 + 320*x^749 + 328*x^750 + 328*x^751 + 328*x^752 + 328*x^753 + 328*x^754 + 328*x^755 + 328*x^756 + 328*x^757 + 328*x^758 + 328*x^759 + 336*x^760 + 336*x^761 + 336*x^762 + 336*x^763 + 336*x^764 + 336*x^765 + 336*x^766 + 336*x^767 + 336*x^768 + 336*x^769 + 344*x^770 + 344*x^771 + 344*x^772 + 344*x^773 + 344*x^774 + 344*x^775 + 344*x^776 + 344*x^777 + 344*x^778 + 344*x^779 + 352*x^780 + 352*x^781 + 352*x^782 + 352*x^783 + 352*x^784 + 352*x^785 + 352*x^786 + 352*x^787 + 352*x^788 + 352*x^789 + 360*x^790 + 360*x^791 + 360*x^792 + 360*x^793 + 360*x^794 + 360*x^795 + 360*x^796 + 360*x^797 + 360*x^798 + 360*x^799 + 369*x^800 + 369*x^801 + 369*x^802 + 369*x^803 + 369*x^804 + 369*x^805 + 369*x^806 + 369*x^807 + 369*x^808 + 369*x^809 + 378*x^810 + 378*x^811 + 378*x^812 + 378*x^813 + 378*x^814 + 378*x^815 + 378*x^816 + 378*x^817 + 378*x^818 + 378*x^819 + 387*x^820 + 387*x^821 + 387*x^822 + 387*x^823 + 387*x^824 + 387*x^825 + 387*x^826 + 387*x^827 + 387*x^828 + 387*x^829 + 396*x^830 + 396*x^831 + 396*x^832 + 396*x^833 + 396*x^834 + 396*x^835 + 396*x^836 + 396*x^837 + 396*x^838 + 396*x^839 + 405*x^840 + 405*x^841 + 405*x^842 + 405*x^843 + 405*x^844 + 405*x^845 + 405*x^846 + 405*x^847 + 405*x^848 + 405*x^849 + 414*x^850 + 414*x^851 + 414*x^852 + 414*x^853 + 414*x^854 + 414*x^855 + 414*x^856 + 414*x^857 + 414*x^858 + 414*x^859 + 423*x^860 + 423*x^861 + 423*x^862 + 423*x^863 + 423*x^864 + 423*x^865 + 423*x^866 + 423*x^867 + 423*x^868 + 423*x^869 + 432*x^870 + 432*x^871 + 432*x^872 + 432*x^873 + 432*x^874 + 432*x^875 + 432*x^876 + 432*x^877 + 432*x^878 + 432*x^879 + 441*x^880 + 441*x^881 + 441*x^882 + 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df (gauss)
gauss
Grup: Administrator
Mesaje: 6933
07 Jan 2011, 03:36

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O solutie mai prozaica, fara sa foloseasca mai mult decat numararea banilor si calculatorul este cea de culoare pytonica...

Calculul este optimal si suficient de rapid..

sage: nr(100)
12
sage: nr(1997)
2650
sage: nr(2011)
2716


Nu doresc prin aceasta sa arat cat de usoara e lumea folosind calculatorul.
Dar pentru astfel de probleme recomand calculatorul.
(Pentru demonstrarea de propozitii calitative legate de functia de partitie recomand desigur calculatorul pentru investigare experimentala si confirmare a formulelor combinatorice intermediare, urmata de demonstratie...)


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df (gauss)
[1]


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