Веленкин;Комбинаторика
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vzrylj wéqéjktqkj¹ wvrjojznsq wéq [1 wvsy·jnu
& (Q 1) = ∑ (− )N &N &1−NP |
= ∑ (− )N &N &Q− |
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Q Q+1−NP− |
Q Q+1−NP− |
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xyuuqévkjtqn knlnzx¹ ls¹ N = lv zn} wvé wvrj k€wvst¹nzx¹ tnéjkntxzkv NP ≤ 1
Æ w u€ éjxxujzéqkjsq ojlj·y v rvsq·nxzkn x·jxzsqk€} iqsnzvk ² y rvzvé€} xyuuj wnék€} zén} |q{é éjktj xyuun wvxsnltq} zén} |q{é cxsq ojuntqz• wvxsnltqn zéq |q{é€ iqsnzj q} lvwvstntq¹uq lv zv wvsy·qzx¹ iqsnz xyuuj |q{é
rvzvévmv éjktj wvëzvuy ·qxsv zjrq} iqsnzvk éjktv & ( ) Ònwné• u€ uv nu tjpzq ëzv ·qxsv wv {véuysn
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bjtv yéjktntqn [ + [ + + [S = P Ëjplqzn ·qxsv nmv |ns€} |
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én¡ntqp ylvksnzkvé¹í•q} tnéjkntxzkju ≤ O ≤ [N ≤ Q P S O Q ² ojljtt€n |ns€n ·qxsj
j Éntzj lsqtvp xu ojréj¡qkjnzx¹ k xsnlyí•nu wvé¹lrn réjxt€p ins€p xqtqp réjxt€p ins€p xqtqp q z l wéq·nu wnék€u qlnz réjxt€p |knz j wvxsnltqu ² xqtqp Çj l€p |knz ojtqujnz kxnmv xu lsqt€ wvsvx tn untnn xu wéq·nu kxn ëzq lsqt€ ² |ns€n ·qxsj frvs•rv kvouv tv xwvxvivk zjrvp éjxréjxrq"
i frvs•rv xy•nxzkynz xwvxvivk zjrvp éjxréjxrq nxsq wvxsnltqu |knzvu uv nz i€z• síivp"
k bvrj qzn ·zv nxsq lsqtj rj lvp wvsvx€ tn untnn xu zv k
xsy·j¹} wvxsnltqu iylnz xqtqp |knz k ² ins€p q k ² réjxt€p
Êunnzx¹ tjivé xvxzjkj U U NUN z n xvxzv¹•qp qo U éjosq·t€} vlqtv·t€} wénlunzvk U wjé éjosq·t€} wénlunzvk U
zévnr q z l bvrj qzn ·zv wévqokvl¹•np {ytr|qnp ls¹ ·qxsj xwvxvivk k€iéjz• Q wénlunzvk qo ëzvmv tjivéj ¹ks¹nzx¹
( − W )U ( − W )U ( − WN+ )UN ( − W)−U −U − −UN
Ëjwq¡qzn wévqokvl¹•yí {ytr|qí ls¹ ·qxsj xwvxvivk k€sv qz• k é¹l Q wénlunzvk qo tjivéj xvxzjkj U U NUN
Êunnzx¹ ëronuws¹évk vltvp rtqmq ² léymvp q ² zénz•np frvs•rquq xwvxvijuq uv tv éjolnsqz• q} un ly síl•uq zjr ·zvi€ rj l€p wvsy·qs rtqm"
Ìivotj·qu ·néno ÍNQ ·qxsv xwvxvivk éjolnsqz• Q éjosq·t€}
wénlunzvk tj N méyww bvrj qzn ·zv − Í + Í − Í + = |
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bvrj qzn |
·zv nxz• V + V + − S + T + U |
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¡jévk un |
ly lkyu¹ sq|juq wéq rvzvé€} rj |
Çjr qountqzx¹ vzknz nxsq T + U < S"
xwvxvivk éjolnsj ëzq} l€p wvsy·jnz V ¡jévk
Ëjplqzn rvsq·nxzkv q ·jxzvzy wv¹ksntq¹ x·jxzsqk€} otj· t€} zjsvtvk tj wévnol
Íévqokvl¹•qn {ytr|qq q tjivé€ mqé•
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rm Ìt wvokvsqs i€ vzkn¡qkjz• síivn |nsvn ·qxsv rqsvméju uvk vz lv rm Íéq ëzvu rj l€p knx wvsy·jsx¹ i€ nlqtxzkntt€u viéjovu
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Ævvi•n qun¹ wv vltvp mqén knxvu k |
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uv tv |
nlqtxzkntt€u |
viéjovu wvsy·qz• |
síivp knx vz lv |
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rm rsjl¹ mqéq tj vlty ·j¡ry knxvk êzv yzkné lntqn |
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uv tv i€sv i€ lvrjojz• qxwvs•ovkjk lkvq·tyí xqxznuy x·q xsntq¹ Ëv vtv v·nt• wévxzv lvrjo€kjnzx¹ q x wvuv••í wév qokvl¹•q} {ytr|qp cxsq k k€éj ntqq
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éjxré€z• xrvirq zv rvë{{q|qntzvu wéq [1 iylnz ·qxsv wv rjo€kjí•nn xrvs•rquq xwvxvijuq uv tv wénlxzjkqz• 1 k
kqln xyuu€ ·qxns Q nxsq rj lvn qo tq} inénzx¹ tn ivsnn vltvmv éjoj Æunxzv zvmv ·zvi€ éjxré€kjz• xrvirq yutv qu I([) tj − [ q kxwvutqu ·zv ( − [)( + [) = − [
( − [ )( + [ ) = − [ q z l Ívëzvuy xknétyk kxn wévqokn lntq¹ u€ wvsy·qu k rvt|n rvt|vk ( − [)I([) = − [ Q+ q wv
zvuy I([) = − [
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− [ Ëv wéjkj¹ ·jxz• ëzvp {véuys€ tn ·zv qtvn
rjr k€éj ntqn ls¹ xyuu€ mnvunzéq·nxrvp wévménxxqq xv otj untjznsnu [ q wnék€u ·sntvu xvxzv¹•np qo Q+ ·sntvk
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
Ívëzvuy
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Ívsy·nttvn éjkntxzkv q wvrjo€kjnz ·zv síivn tjzyéjs•tvn ·qxsv vz lv Q+ − uv tv nlqtxzkntt€u viéjovu wénlxzj
kqz• k kqln xyuu€ ·qxns Q iné¹ q} tn ivsnn vltvmv éjoj
f jsmniéjq·nxrvp zv·rq oéntq¹ wévknlnttvn lvrjojzns•xzkv vxtvk€kjsvx• tj zvu ·zv utvmv·snt + [ + [ + [ + + [N
wéq N = Q+ − lvwyxrjnz |
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ntqn |
tj utv qznsq |
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rjrvu tqiyl• |
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Êo kxn} tjivévk xv xkvpxzkvu nlqtxzknttvxzq lkvq·t€p
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wénlxzjkqz• nlqtxzkntt€u viéjovu k kqln xyuu€ tnrvzvé€} ·qxns qo tjivéj ± ± ± Q iné¹ rj lvn ·qxsv tn ivsnn
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um q z l uv tv xvxzjkqz• síivp knx k€éj ntt€p |ns€u ·qxsvu uqssqméjuuvk
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tv zjr éjxrvkjz• Q oknt•nk ëzvp |nwq tv iylnz tjiéjz• síiyí ujxxy vz lv
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bvrj qzn ·zv wéq P > Q
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qzn ·zv (& ) + (& ) + (& ) + + Q(&Q) = |
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èjxxujzéqkj¹ k€éj ntqn |
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>( + [)S+ ± ( − [)S+ @ >( + [)S ± ( − [)S@ |
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wéq kxn} rvuiqtj|q¹} otjrvk k€knlqzn {véuys€ |
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-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
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V P− V+ = & P+ − (− )P&P |
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rj lj¹ |
xyuuj éjxwévxzéjtntj tj |ns€n tnvzéq|jzns•t€n otj·ntq¹ |
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V ls¹ rvzvé€} vwénlnsntj snkj¹ ·jxz• éjkntxzkj |
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èjxxuvzénk iqtvu |
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lvrj |
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j − & + & − & + = (− )Q QFRV Qπ |
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& − & + |
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(− )Q+ |
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Íyxz• ε = FRV π |
+ L VLQ π |
èjxxuvzénk iqtvu ( + [)Q wvxsnlv |
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kjzns•tv wéq [ = [ = ε [ = ε |
q [ = L lvrj |
qzn ·zv |
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j |
& + & + & + = |
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& + & + & + = |
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Q− |
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+ FRV Q |
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j Êo zv |
lnxzkj ( + [)S( + [)−N− = ( + [)S−N− k€knlqzn ·zv |
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∑ (− )V&V |
Q−V = &Q |
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N+V&S |
S−N− olnx• q ljsnn xyuuj éjxwévxzéjtntj tj |ns€n |
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tnvzéq|jzns•t€n otj·ntq¹ V ls¹ rvzvé€} vwénlnsntj snkj¹ ·jxz• éjkntxzkj
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djlj·q |
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i Êo zv |
lnxzkj ( − [)−P− ( − [)−T− = ( − [)−P−T− k€knlqzn ·zv |
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k Êo zv |
lnxzkj ( + [)Q = ( − [ )Q( − [)−Q k€knlqzn ·zv |
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∑ (− )V&Q |
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m Êo zv |
lnxzkj ( + [)Q( − [ )−Q = ( − [)−Q k€knlqzn ·zv |
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∑ &NQ− V |
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l Êo zv |
lnxzkj ( − [ )−S− = ( + [)−S− ( − [)−S− k€knlqzn ·zv |
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= &N |
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S+ N−V&S+V |
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S+N |
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V
Êo zv lnxzk N = − S k€knlqzn ·zv
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= ( − [)N ls¹ N = S q |
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bvrj |
qzn ·zv ∑ |
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S+ V = P− S + P |
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&S+V& S+P |
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Êo zv lnxzk ( − |
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= ( + [ )N |
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ls¹ N = S q |
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N = − S k€knlqzn {véuys€ |
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( − |
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∑ (− )V&V |
P+ −V |
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∑ (− )V&V |
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P−V |
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S+V− & P+ S+V |
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S+V− & P+ S+V− |
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∑ (− )V&V& P+ −V V = |
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∑ (− )V&V& P−V V = &P |
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S S− V |
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S S− V |
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f q} wvuv••í lvrj qzn ·zv |
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= P (S + P) |
(S + P − ) |
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V |
& S+ P&S+P−V |
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S ( P) |
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S ( P + ) |
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V |
& S+ P+ &S+P−V |
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= P− |
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& S+ P&S+P−V |
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&S+ P− |
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& S+ P+ &S+P−V |
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èjxxujzéqkj¹ {véuys€
>( + [)S ± ( − [)S@ = ( + [) S + ( − [) S ± ( − [ )S
>( + [)S + ( − [)S@ >( + [)S − ( − [)S@ = ( + [) S − ( − [) S
-sjkj 9,,, è‰bÔ Ê ÍèÌÊdÆÌb‰ÕÊc ÐÏËÇgÊÊ
wéq wvsv |
qzns•t€} q vzéq|jzns•t€} otj·ntq¹} S lvrj qzn ·zv |
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∑ |
&SV&SP− V = & P + (− ) |
P&P |
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&SV&SP− V+ = |
& P+ |
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&SV+ P− V+ |
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&S |
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+ (− ) |
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&S+ V&S+ P− V |
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S− S− |
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Êo xvvztv¡ntq¹ |
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k€knlq |
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zn ·zv ∑ |
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(− )V&P−N+V |
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nxsq |
P ≠ Q |
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bvrj qzn ·zv ∑ (− )V&V &Q = |
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P = Q |
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éjkntxzkj |
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( − [)−Q ( − [K)Q = ( + [ + + [K− )Q |
k€knlqzn |
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nxsq |
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·zv ∑ (− )V& |
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Êo |
zv |
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lnxzkj |
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( − [)−Q− ( − [K)Q = |
( + [ + + [K− )Q |
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zn ·zv ∑ |
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Êo zv |
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lnxzk |
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( + [)N ( − [)N = ( − [ )N |
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k€knlqzn ·zv |
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(− ) |
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