Веленкин;Комбинаторика
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-sjkj ,, и¬dОcХcЛК‰ НcиcfТ¬ЛМЖЗК К fМпcТ¬ЛК‰
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n kv kzvévp xzévrn ësnuntzvk zqwj 1(DLDM) iylnz |
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& wvëzvuy 1(D D ) + + 1(D |
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&Q1 Ívëzvuy k éjxxujzéqkjnuvu xsy·jn {véuysj krsí·ntqp |
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dtjrvwnénuntt€n xyuu€ xv·nzjtqp
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Íévxznp¡np qo zjrq} {véuys ¹ks¹nzx¹
& − & + & + + (− )Q&Q = |
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êzv zv lnxzkv xéjoy k€znrjnz qo éjkntxzkj ² xyuuj xv·nzjtqp wv ·nztvuy ·qxsy ësnuntzvk éjktj xyuun xv·nzjtqp wv tn·nztvuy ·qxsy ësnuntzvk
êzy {véuysy uv tv lvrjojz• rvuiqtjzvétv Æ€wq¡nu kxn xv·nzj tq¹ qo Q ësnuntzvk D DQ q xlnsjnu xsnlyí•nn wénviéjovkjtqn
r xv·nzjtqí tn xvlné j•nuy iyrky D wéqwq¡nu ëzy iyrky j qo
xv·nzjtqp rylj vtj k}vlqz k€·nértnu nn Énmrv wévknéqz• ·zv wéq ëzvu xtvkj wvsy·jzx¹ kxn xv·nzjtq¹ q wéqzvu wv vltvuy éjoy Ëv wéq ëzvu wénviéjovkjtqq kxn xv·nzjtq¹ qunk¡qn ·nztvn ·qxsv ësnuntzvk wénkéjz¹zx¹ k xv·nzjtq¹ quní•qn tn·nztvn ·qxsv ësnuntzvk q viéjztv dtj·qz xv·nzjtqp x ·nzt€u ·qxsvu ësnuntzvk xzvs•rv n xrvs•rv q x tn·nzt€u ·qxsvu u€ krsí·jnu k éjxxuvzéntqn q wyxzvn xv·nzjtqn xvkxnu tn xvlné j•nn ësnuntzvk êzv q k€éj jnzx¹ {vé uysvp
bvrj nu wéq P > ivsnn xsv tyí {véuysy
& &P − & &P− + & &P− − + (− )P&P & |
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Q Q− Q Q− |
Q Q−P |
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bs¹ lvrjojzns•xzkj éjxxuvzéqu xv·nzjtq¹ qo Q ësnuntzvk D DQ wv P Ìivotj·qu ·néno D DN xkvpxzkv xv·nzjtq¹
ojrsí·jí•nnx¹ k zvu ·zv k ëzv xv·nzjtqn ojknlvuv k}vl¹z ësnuntz€ D DN ïqxsv 1(D DN) zjrq} xv·nzjtqp éjktv
k tq} N unxz ojt¹z€ ësnuntzjuq D DN j tj vxzjk¡qnx¹ P − N unxz nxz• Q − N wénzntlntzvk Ìi•nn ·qxsv
dtjrvwnénuntt€n xyuu€ xv·nzjtqp
xv·nzjtqp éjktv &P ëzv |
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wvsy·jnzx¹ wéq N = j |
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xv·nzjtqp tn visjljí•q} tq vltqu qo xkvpxzk D DQ |
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wéq P > tn xy•nxzkynz k rj |
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k}vl¹z rjrqn zv |
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lvn xv·nzjtqn _ _ |
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ësnuntz€ |
Ívëzvuy |
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tj¡nu |
xsy·jn |
1(D D D DQ) = |
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1 |
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= &P−N Ívlxzjks¹¹ ëzq otj·ntq¹ k {véuysy wéq}vlqu |
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Q−N |
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r zv |
lnxzky |
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Òv·tv zjr |
n lvrjo€kjízx¹ xvvztv¡ntq¹ |
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− & &P− |
+ & &P− |
− + (− )Q&Q &P−Q |
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&Q &Q+P− |
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Q Q+P− |
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Q Q+P− |
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Q P− |
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nxsq P ≥ Q |
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+ & &P− |
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&Q &Q+P− |
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Q Q+P− |
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Q Q+P− |
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Q Q− |
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nxsq P < Q |
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Êunttv |
éjxxuvzéqu xv·nzjtq¹ |
x wvkzvéntq¹uq qo |
Q ësn |
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untzvk D D DQ |
wv P q vivotj·qu ·néno DN ≤ N ≤ Q |
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xkvpxzkv xv·nzjtq¹ ojrsí·jí•nnx¹ k zvu ·zv xénlq nmv ësn |
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untzvk nxz• |
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nz léymqn ësnuntz€ |
Òvmlj |
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1(D DN) ² ·qxsv xv·nzjtqp k rvzvé€n ojknlvuv k}vl¹z |
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ësnuntz€ D DN |
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Êo |
rj lvmv |
zjrvmv xv·nzjtq¹ |
uv tv |
yljsqz• wv |
vltvuy |
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ësnuntzy D DN Æ énoys•zjzn wvsy·qzx¹ tnrvzvévn xv·nzjtqn x wvkzvéntq¹uq qo Q ësnuntzvk D DQ wv P − N Íéq ëzvu
viéjztv lvijks¹¹ r xv·nzjtqí x wvkzvéntq¹uq qo ësnuntzvk
D DQ xvlné j•nuy P − N ësnuntzvk wv vltvuy ësnuntzy |
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D DN wvsy·qu xv·nzjtqn xvlné j•nn P ësnuntzvk k |
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rvzvévu ojknlvuv wénlxzjksnt€ D DN Ìzxílj k€znrjnz |
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·zv ·qxsv 1(D DN) éjktv ·qxsy xv·nzjtqp x wvkzvéntq¹uq qo |
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Q ësnuntzvk wv P − N z n 1(D D ) = &P−N |
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Q+P−N− |
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bjsnn vi•nn ·qxsv xv·nzjtqp x wvkzvéntq¹uq qo Q ësnunt
P
zvk D DQ wv P éjktv &Q+P− ëzv 1 j xv·nzjtqp tn
visjljí•q} tq vltqu qo xkvpxzk D ≤ N ≤ Q tn xy•nxzkynz |
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Ívlxzjks¹¹ |
tjplntt€n |
otj·ntq¹ |
1(D D D DQ) = |
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1N = &QP+−PN−N− |
k {véuysy wéq}vlqu r zv lnxzkju |
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-sjkj ,, и¬dОcХcЛК‰ НcиcfТ¬ЛМЖЗК К fМпcТ¬ЛК‰
Ëjrvtn| lvrj nu wéq P < Q zv lnxzkv
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& QP − & (Q − )P + & (Q − )P − + (− )Q− &Q− P = |
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bs¹ ëzvmv éjxxuvzéqu éjoun•ntq¹ x wvkzvéntq¹uq |
qo Q |
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ësnuntzvk wv P q vivotj·qu ·néno DN xkvpxzkv éjoun•ntq¹ |
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xvxzv¹•nn k zvu ·zv k tnmv tn k}vl¹z ësnuntz€ zqwj |
D Òvmlj |
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1(D DN) ² ·qxsv éjoun•ntqp x wvkzvéntq¹uq xvlné |
j•q} |
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P ësnuntzvk xénlq |
rvzvé€} tnz ësnuntzvk D |
DN |
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k€iqéjz• wéq}vlqzx¹ sq¡• qo Q − N ësnuntzvk DN+ DN+ DQ |
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éjoun•ntqp |
éjktv (Q − N)P |
zjr |
·zv |
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1(D DN) = (Q − N)P Ìi•nn ·qxsv éjoun•ntqp éjktv QP |
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Ëjrvtn| éjoun•ntqp tn visjljí•q} tq vltqu qo xkvpxzk |
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D tn xy•nxzkynz Æ |
xjuvu lnsn nxsq éjoun•ntqn |
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tn visjljnz tq vltqu qo xkvpxzk DN zv vtv xvlné |
qz kxn Q |
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ësnuntzvk Ëv u€ wénlwvsjmjnu ·zv ·qxsv P_ ësnuntzvk k éjoun•ntqq unt•¡n ·nu Q Ívëzvuy 1(D DQ) = q u€
wéq}vlqu r zv lnxzky
΀ lvrjojsq olnx• |ns€p é¹l xvvztv¡ntqp ls¹ ·qxns &NQ
Ê} uv tv lvrjojz• q léymquq xwvxvijuq Æ w u€ éjxxrj nu v mnvunzéq·nxrvu xwvxvin lvrjojzns•xzkj zjrq} xvvztv¡n tqp j k msjkn 9,,, qosv qu xju€p xqs•t€p unzvl lvrjojzns• xzkj ² unzvl wévqokvl¹•q} {ytr|qp êzqu unzvlvu uv tv lvrjojz• tn zvs•rv kxn xvvztv¡ntq¹ wéqknlntt€n k ëzvp msjkn
tv q |ns€p é¹l léymq} qtznénxt€} xvvztv¡ntqp
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nsnotvlvév |
tvmv kjmvtj qunnzx¹ lkj wévzqkvwvsv |
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t€} lqkjtj wv unxz k rj |
lvu Êo wjxxj |
qévk ·nzknév nsjíz |
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xqlnz• wv lkq |
ntqí wvnolj zévn ² wévzqk lkq |
ntq¹ vxzjs•t€u zénu |
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inoéjosq·tv rjr xqlnz• frvs•rquq xwvxvijuq uvmyz éjounxzqz•x¹ wjxxj qé€"
Êunnzx¹ Q jivtntzvk znsn{vttvp xnzq frvs•rquq xwvxvijuq uv tv vltvkénunttv xvnlqtqz• zéq wjé€ jivtntzvk"
Ëj zjt|€ wéq¡sj rvuwjtq¹ qo ítv¡np q lnky¡nr
j frvs•rv qunnzx¹ kjéqjtzvk y·jxzq¹ lnky¡nr k zjt|n k rvzvévu zjt|yíz kxn ítv¡q"
i frvs•rv qunnzx¹ kjéqjtzvk nxsq y·qz€kjz• sq¡• zv rjrqn lnky¡rq vxzjsqx• tnwéqmsj¡ntt€uq"
k Ìzknz•zn tj zn n kvwévx€ nxsq vztvxqzns•tv lky} lnky¡nr uv tv x yknénttvxz•í xrjojz• ·zv vtq iylyz wéqmsj¡nt€ tj zjtn|
Ëj ¡rvs•tvu kn·nén wéqxyzxzkyíz lnky¡nr q ítv¡np frvs•rquq xwvxvijuq uv tv k€iéjz• qo tq} wjé€ ls¹ zjt|j"
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Æv xrvs•rq} ¡nxzqotj·t€} ·qxsj} nxz• ·nzt€n q tn·nzt€n |q{é€ nxsq lvwyxrjízx¹ q ©¡nxzqotj·t€nª ·qxsj tj·qtjí•qnx¹ x tys¹" ¬ nxsq tn lvwyxrjízx¹"
Æv xrvs•rq} lnx¹zqotj·t€} ·qxsj} xyuuj |q{é éjktjwnékj¹ |q{éj wénlwvsjmjnzx¹ vzsq·tvp vz " ¬ nxsq inéyzx¹ tn lnx¹zqotj·t€n ·qxsj j kxn ·qxsj vz lv krsí·qzns• tv"
frvs•rv ·qxns vz lv xvlné qz |q{éy " frvs•rv ·qxns xvlné qz nn lkj l€"
frvs•rv ·qxns xvlné qz |q{éy " frvs•rv ·qxns xvlné qz lkj l€"
frvs•rv ·qxns xvlné qz |q{é€ q " ¬ |q{é€ q "
Ï Íj¡q léyonp Æ zn·ntqn tnlnsq vt wéqmsj¡jnz q} r xnin wv vinljz• wéq·nu rvuwjtqq tn wvkzvé¹ízx¹ frvs•rquq xwvxvijuq vt uv nz xvxzjkqz• éjxwqxjtqn vinlvk zjr ·zvi€
j tqrjrqn lkj léymj tn kxzénzqsqx• y tnmv ivsnn vltvmv éjoj i tqrzv tn vxzjsx¹ tnwéqmsj¡ntt€u
k tq vlqt qo léyonp tn wvxn•js nmv rj l€p lnt•"
Ènéyzx¹ rvxzq ©vivi•nttvmvª lvuqtv vz _ lv Q _ Q bvrj qzn ·zv ·qxsv rvxznp x xyuuvp v·rvk Q − U éjktv ·qxsy rvxznp x
xyuuvp v·rvk Q + U Ëjplqzn vi•nn ·qxsv kxn} rvxznp zjrvmv lvuqtv
frvs•rquq xwvxvijuq uv tv k€iéjz• qo ·nsvknr méywwy sílnp ls¹ éjivz€ nxsq k méywwy uvmyz k}vlqz• vz lv ·nsvknr"
¬nxsq méywwj lvs tj xvxzv¹z• tn untnn ·nu qo ·nsvknr"
¬nxsq ·qxsv sílnp Q j tjqunt•¡nn ·qxsv sílnp k méywwn S"
frvs•rquq xwvxvijuq uv tv k€iéjz• ·nsvknr qo nxsq
xénlq tq} nxz• lkvn rvzvé€} tns•o¹ k€iqéjz• kunxzn"
frvs•rquq xwvxvijuq uv tv k€iéjz• P ·nsvknr qo Q zjr ·zvi€ ljtt€n S ·nsvknr tn i€sq k€iéjt€ kunxzn"
Ï uy j otjrvu€} ² nt•qt q uy ·qt j y nt€ ²
nt•qt q uy ·qt qt€n ·nu y uy j frvs•rquq xwvxvijuq
uv tv xvxzjkqz• rvuwjtqí qo uy ·qt q nt•qt zjr ·zvi€ ·nsvknr wéqmsjxqs uy q ² ntj"
Ëj rj lvu ivézy svlrq y knxns lvs t€ xqlnz• wv ·nsvknrj frvs•rquq xwvxvijuq uv tv k€iéjz• rvujtly ls¹ ëzvp svlrq nxsq nxz• rjtlqljz wéq·nu ·nsvknr }vz¹z xqlnz• tj snkvu ivézy svlrq ² tj wéjkvu j ls¹ inoéjosq·tv mln xqlnz•"
Æ yétn sn jz nzvt€ x ·qxsjuq
Êo tnn k€tqujíz nzvtj Æv xrvs•rq} xsy·j¹} xyuuj tjwqxjtt€} tj tq} ·qxns iylnz éjktj " Ën unt•¡n "
frvs•rv éjosq·t€} lnx¹zqotj·t€} ·qxns uv tv tjwqxjz• wvs•oy¹x• sq¡• |q{éjuq wéq lvwvstqzns•tvu yxsvkqq ·zv |q{éj qxwvs•oynzx¹ k rj lvu ·qxsn évktv lkj éjoj" frvs•rv qo tjwqxjtt€} ·qxns lnsqzx¹ tj "
Êo lnky¡nr q ítv¡np k€iqéjíz rvujtly k xvxzjkn ·nsvknr frvs•rquq xwvxvijuq uv tv k€iéjz• ëzy rvujtly zjr ·zvi€ k tnn kv¡sv tn ivsnn ítv¡np"
Êunnzx¹ ins€} ¡jéj xqtq} q réjxt€} frvs•rquq xwvxvijuq uv tv k€sv qz• qo tq} é¹l qo Q ¡jévk nxsq Q =
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-sjkj ,, и¬dОcХcЛК‰ НcиcfТ¬ЛМЖЗК К fМпcТ¬ЛК‰
frvs•rquq xwvxvijuq uv tv qoksn·• qo rvsvl€ k rjéz€rjéz zjr ·zvi€
j vtq i€sq vltvp ujxzq q ¡sq wvlé¹l tjwéquné zyo lkvprj zévprj ·nzknérj w¹znérj qsq lnx¹zrj kjsnz ljuj rvévs• zyo
i vtq i€sq vltvp ujxzq
k vtq ¡sq wvlé¹l tjwéquné zyo lkvprj zévprj ·nzknérj w¹znérj qsq lnx¹zrj kjsnz ljuj rvévs• zyo "
j frvs•rv uv tv xvxzjkqz• w¹zqiyrkntt€} xsvk qo msjxt€} q xvmsjxt€} iyrk iyrk€ uvmyz wvkzvé¹z•x¹ nxsq msjxt€n q xvmsjxt€n iyrk€ lvs t€ ·nénlvkjz•x¹"
i frvs•rv qo ëzq} xsvk tj·qtjízx¹ msjxtvp iyrkvp q xrvs•rv xvmsjxtvp"
k èn¡qzn zn n ojlj·q nxsq tq vltj iyrkj tn wvkzvé¹nzx¹ lkj l€
Ìémrvuqznz vsquwqjl€ xvxzvqz qo ·nsvknr Îjznéqjs€ vsquwqjl€ }éjt¹zx¹ k xnp{n frvs•rv ojurvk lvs nt qunz• xnp{ q xrvs•rquq rsí·juq xsnlynz xtjilqz• rj lvmv ·sntj vémrvuqznzj ·zvi€ lvxzyw k xnp{ i€s kvouv nt rvmlj xvinéyzx¹ síi€n ·sntvk vémrvuqznzj q tn i€s kvouv nt nxsq xvinéyzx¹ unt•¡n ·sntvk"
j frvs•rquq éjosq·t€uq xwvxvijuq uv tv k€iéjz• tnxrvs• rv iyrk qo {éjo€ ©Ìrv oj vrv oyi oj oyiª" Ívé¹lvr iyrk tn y·qz€kjnzx¹
i frvs•rquq xwvxvijuq uv tv k€iéjz• qo zvp n {éjo€ zéq iyrk€"
k frvs•rquq xwvxvijuq uv tv k€iéjz• qo ëzvp {éjo€ zéq iyrk€ nxsq y·qz€kjz• wvé¹lvr k€iéjtt€} iyrk"
Ëj wsvxrvxzq ljt€ zv·nr wéq·nu xénlq wé¹u€} xvnlqt¹ í•q} ëzq zv·rq tnz wjéjssns•t€} wnéwntlqrys¹ét€} qsq xvkwjlj í•q} Íévkvlqu ·néno rj lyí qo zv·nr wnéwntlqrys¹é€ rv kxnu wé¹u€u rvzvé€n uv tv wvxzévqz• xvnlqt¹¹ léymqn zv·rq Çjrvkv ujrxqujs•tvn ·qxsv zv·nr wnénxn·ntq¹ un ly xvivp ëzq} wnéwntlq rys¹évk tn x·qzj¹ ljtt€n zv·nr"
frvs•rquq xwvxvijuq uv tv k€iéjz• qo sv¡jlnp ¡nxznéry ls¹ ojwé¹ rq zjr ·zvi€ k tnn kv¡sq sv¡jlq qo ¡nxznérq $%&$′%′&′ tv tn kv¡sj tq vltj qo wjé $$′ %%′ &&′"
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uv tv xvxzjkqz• qo |
xvmsjxt€} |
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k}vl¹z éjosq·t€} |
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Æv xrvs•rq} qo ëzq} xsvk tqrjrqn lkn xvmsjxt€n tn xzv¹z é¹lvu"
Ëjplqzn rvsq·nxzkv ¡nxzqotj·t€} ·qxns zjrq} ·zv xyuuj zén}otj·tvmv ·qxsj viéjovkjttvmv wnék€uq zénu¹ |q{éjuq q zén} otj·tvmv ·qxsj viéjovkjttvmv wvxsnltquq zénu¹ |q{éjuq unt•¡n
frvs•rquq xwvxvijuq uv tv wnénxzjkqz• iyrk€ xsvkj ©rv{n kjérjª zjr ·zvi€ msjxt€n q xvmsjxt€n iyrk€ ·nénlvkjsqx•" ¬ k xsvkn ©xjuvkjéª"
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frvs•rv tnvzéq|jzns•t€} |ns€} ·qxns unt•¡q}
xvlné jz kxn |q{é€ "
frvs•rv ·qxns xvxzvqz zvs•rv qo ëzq} |q{é"
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frvs•rquq xwvxvijuq uv tv éjxxzjkqz• ins€} q ·nét€} ¡j¡nr tj ·nét€} wvs¹} ¡j}ujztvp lvxrq"
frvs•rquq xwvxvijuq uv tv k€iéjz• qo xsvkj ©svmjéq{uª lkn xvmsjxt€} q vlty msjxtyí iyrky"
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