COMBINATORIAL MINORS FOR MATRIX FUNCTIONS AND THEIR APPLICATIONS

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1 ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2014 Seria: MATEMATYKA STOSOWANA z. 4 Nr kol Vladimir SHEVELEV Department of Mathematics Ben-Gurion University of the Negev COMBINATORIAL MINORS FOR MATRIX FUNCTIONS AND THEIR APPLICATIONS Summary. As well known, permanent of a square(0,1)-matrix A of ordernenumeratesthepermutationsβof1,2,...,nwiththeincidencematricesb A.ToobtainenumerativeinformationonevenandoddpermutationswithconditionB A,weshouldcalculatetwo-foldvector(a 1,a 2 ) witha 1 +a 2 =pera.moregeneral,theintroducedω-permanent,where ω=e 2πi/m,wecalculateasm-foldvector.Fortheseandothermatrixfunctions we generalize the Laplace theorem of their expansion over elements of the first row, using the defined so-called combinatorial minors. In particular, in this way, we calculate the cycle index of permutations with condition B A. MINORY KOMBINATORYCZNE DLA FUNKCJI MACIERZOWYCH I ICH ZASTOSOWANIA Streszczenie. Jak wiadomo, permanent(0, 1)-macierzy kwadratowej Astopnianpodajeliczbępermutacjiβliczb1,2,...,n,mającychmacierz incydencjib A.AbyotrzymaćinformacjęoliczbieparzystychinieparzystychpermutacjizwarunkiemB A,należyobliczyćdwuskładowywektor(a 1,a 2 ),gdziea 1 +a 2 = pera.ogólniejwprowadzamypo Mathematics Subject Classification: 15A15. Keywords: square matrices, permanent, cycle permanent, ω-permanent, cycle index of permutations with restricted positions. Corresponding author: V. Shevelev(shevelev@bgu.ac.il). Received:

2 6 V. Shevelev jęcieω-permanentu,gdzieω=e 2πi/m,któryobliczamyjakoodpowiedni m-składowy wektor. Dla takich i innych funkcji macierzowych uogólniamy twierdzenie Laplace a o ich rozwinięciu względem elementów pierwszego wiersza, wykorzystując zdefiniowane w tym celu tak zwane minory kombinatoryczne. W szczególności obliczamy w ten sposób indeks cyklowy permutacjispełniającychwarunekb A. 1. Introduction LetS (n) besymmetricgroupofpermutationsofnumbers{1,...,n}.leta= {a ij }besquarematrixofordern.recallthatthepermanentofaisdefinedby formula([4]): pera= n a i,s(i). (1) s S (n) i=1 IfAisa(0,1)matrix,thenitdefinesaclassB=B(A)ofpermutationswith restricted positions, such that the positions of its zeros are prohibited. Such class couldbeequivalentlydefinedbyasimpleinequality:apermutationπ Bifand onlyifforitsincidencematrixp wehavep A.Oneofthemostimportant applicationofperaconsistsoftheequality B =pera.thusperaenumerates permutations with restricted positions of the class B(A). Letγ(π)benumberofindependentcyclesofπ,includingcyclesoflength1. Thenthedifferenced(π)=n γ(π)iscalleddecrement ofπ([3]).permutation πiscalledeven(odd), ifd(π)iseven(odd).notethatthedeterminantofmatrix Acouldbedefinedbyformula n deta= a i,s(i) n a i,s(i). (2) Since, evidently, we also have pera= evens S (n) i=1 evens S (n) i=1 odds S (n) i=1 n a i,s(i) + odds S (n) i=1 n a i,s(i), (3) thenthenumbersofevenandoddpermutationsofclassb(a)aregivenbyvector ( 1 2 (pera+deta), 1 ) 2 (pera deta). (4) Note that, in contrast to permanent, there exist methods of very fast calculation of det A. Therefore, the enumerative information given by(4) one can obtain approximatelyforthesametimeasthenumber B(A) givenby(1).

3 Combinatorial minors for matrix functions and their applications 7 Letm 3and0 k<mbegivenintegers.wesaythatapermutationπ belongstoclasskmodulom(π S (n) k,m ),ifd(π) k(modm).thefirstproblem under our consideration in this paper is the enumeration of permutations π B(A) ofclasskmodulom.itisclearthatthisproblemisanaturalgeneralizationof problem of enumeration of even and odd permutation with restricted positions whichissolvedby(4).inordertosolvethismoregeneralproblem,putω=e 2πi m and introduce a new matrix function which we call ω-permanent. Definition1.LetAbeasquarematrixofordern.Wecallω-permanentofmatrix A the following matrix function per ω A= s S (n) i=1 0,m n a i,s(i) +ω +ω 2 s S (n) i=1 2,m s S (n) i=1 1,m n a i,s(i) + n a i,s(i) +...+ω m 1 s S (n) i=1 m 1,m n a i,s(i). (5) Notethat,ifm=1,thenper ω A=perA,andifm=2,thenper ω A=detA. Incasem 3,everysumin(5)essentiallydiffersfrompermanent.Therefore,the known methods of evaluation of permanent([4, ch. 7]) are not applicable. However, usingso-called combinatorialminors,belowwefindanexpansionper ω Aover thefirstrowofmatrixa.thisallowstoreduceaproblemoforderntoafew problemsofordern 1. The second problem under our consideration is another important problem of enumeration of full cycles of length n with restricted positions. In connection with this problem, we introduce another new matrix function which we call cyclic permanent. Definition2.LetAbesquarematrixofordern.Thenumber Cycl(A)= s n a i,s(i), (6) i=1 wherethesummingisoverallfullcyclesfroms (n),wecallacyclepermanent ofa. The third problem is a problems of enumeration of permutations with restricted positions with a restrictions on their cycle structure.

4 8 V. Shevelev Denote γ(s) the number of all cycles of permutation s(including cycles of length 1, if they exist). Recall([1]) that the absolute value of Stirling number S(n,k)ofthefirstkindequalstonumberofpermutationss S (n) withγ(s)= k([1,5]).anaturalgeneralizationofstirlingnumbersofthefirstkindisthe following matrix function. Definition 3. The matrix function n S(A;n,k)= a i,s(i), (7) s S (n),γ(s)=ki=1 wherenisorderofsquarematrixa,wecallstirlingfunctionofindexk. Finally,recall([5])thatapermutations S (n) withk 1 cyclesoflength1,k 2 cyclesoflength2,andsoon,issaidtobeofcyclestructurek=(k 1,k 2,...,k n ). Denoteν(k 1,k 2,...,k n )thenumberofpermutationsofclassk=(k 1,k 2,...,k n ). Then the polynomial C(t 1,t 2,...,t n )= ν(k 1,k 2,...,k n )t k1 1 tk2 2...tkn n (8) iscalledthecycleindexofpermutationsofs (n).anaturalgeneralizationofthe cycle index on permutations with restricted positions is C(A;t 1,t 2,...,t n )= ν(a;k 1,k 2,...,k n )t k1 1 tk2 2...tkn n (9) whereν(a;k 1,k 2,...,k n )isthenumberofpermutationsofclassb(a)withthe cyclestructurek.inparticular,ifa=j n,wherej n isn nmatrixfrom1 sonly, wehavec(a;t 1,t 2,...,t n )=C(t 1,t 2,...,t n ). 2.Observationsincasem=2ofω-permanent (determinant) Thecaseofdeterminant(m=2)ofω-permanentisauniquecasewhenit is easy to obtain a required enumerative information formally using formulas(4). Forthepassagetoageneralcase,itisimportantforustounderstandhowonecan obtain such information from the definition(2) of determinant only. Essentially, the required information is contained in vector ( n deta= a i,s(i), n a i,s(i) ), (10) even s Si=1 odd s Si=1

5 Combinatorial minors for matrix functions and their applications 9 anditimmediatelydisappears,iftoadditscomponents,orformallytousean identityoftheform1 1=0.Sincea i,j =0or1,theninsumsof(10)we do not use such an identity. However, none of algorithms of fast calculation of determinant exists without using of it. On the other hand, using the Laplace algorithmofexpansionofdeterminantover(thefirst)row,itcanonlybeusedat thelaststep.therefore,ifnottodo thelaststep,wecanobtaintherequired enumerative information. Example 4. By the Laplace expansion, we have det = (1 1)+(1 1)= = and,ifnottodotheuseless(withtheenumerativepointofview)laststep,then we have det =(2, 2) Thismeansthattherearetwoevenandtwooddpermutationswiththeprohibited position(1, 1). Of course, in general, the Laplace expansion of determinant over the first row in its classic form deta= ( 1) j 1 a 1,j M 1j, (11) j=1 wherem 1j isminorofelementa 1j,i.e.,determinantofthecomplementarytoa 1j submatrixa 1j,doesnotworkforω-permanent. Therefore, let us introduce for our aims a more suitable notion of so-called combinatorialminor ofelementa ij.letthecomplementarytoa ij submatrix A ij havethefollowingn 1columns c 1,c 2,...,c j 1,c j+1,...,c n. (12) Thefirstj 1ofthesecolumnswechangeinthefollowingcyclicorder:c 2,c 3,..., c j 1,c 1.ThenweobtainanewmatrixA ijwiththecolumns c 2,c 3,...,c j 1,c 1,c j+1,...,c n. (13) DeterminantofmatrixA ijwecall combinatorialminor(cm) ij ofelementa ij. Itiseasytoseethat (CM) i1 =M i1 ; (CM) ij =( 1) j 2 M ij, j=2,...,n. (14)

6 10 V. Shevelev Therefore, e.g., expansion(11) one can rewrite in the form deta=a 1,1 (CM) 11 a 1,j (CM) 1j. (15) In general, let us give a definition of combinatorial minors for arbitrary matrix function X(A). j=2 Definition5.LetXbematrixfunctiondefinedonallsquarematricesoforder n 3.LetA={a ij }beasquarematrixofordernanda ij bethecomplementary toa ij submatrixwithcolumns(12).denotea ijanewsquarematrixoforder n 1withcolumns(13).ThenthenumberX(A ij)iscalledacombinatorialminor ofa ij. It appears that our observation(15) has a general character. So, in Sections 4 wegiveageneralizationofthelaplaceexpansionoftype(15)forper ω A,Cycl(A), S(A;n,k)andC n (A;t 1,t 2,...,t n ). 3. Main lemma Lemma6.Letπ S (n) withπ(j)=1andσ=σ j (π), j 2,suchthat σ(1)=π(2), σ(2)=π(3),..., σ(j 2)=π(j 1), σ(j 1)=π(1), σ(j)=π(j)=1, σ(j+1)=π(j+1),..., σ(n)=π(n). (16) Let,further,π S (n 1) definedbytheformula π σ(i) 1, if1 i j 1, (i)= σ(i+1) 1, ifj i n 1. (17) Thenpermutationsπandπ havethesamenumberofcycles: γ(π)=γ(π ). (18) Proof. From(16) (17) we find π π(i+1) 1, if i j 1, (i)= π(1) 1, if i=j 1. (19)

7 Combinatorial minors for matrix functions and their applications 11 Consideracycleofπcontainingelementj.Letithaslengthl 2,suchthat π(j)=1, π(1)=k 1, π(k 1 )=k 2,..., π(k l 3 )=k l 2, π(k l 2 )=j. Beginningwiththeequalityπ(1)=k 1,by(19),thismeansthat π (j 1)=k 1 1, π (k 1 1)=k 2 1,..., π (k l 3 1)=k l 2 1, π (k l 2 1)=j 1. Thustoacycleofπoflengthl 2containingelementjcorrespondsacycleof lengthl 1ofπ.Quiteanalogously,weverifythattocycleofπoflengthl 2not containingelementjcorrespondsacycleofthesamelengthofπ.forexample, tocycleoflengthl 2oftheform π(j 1)=k 1, π(k 1 )=k 2, π(k 2 )=k 3,..., π(k l 2 )=k l 1, π(k l 1 )=j 1 (beginningwithπ(k 1 )=k 2 ),correspondsthecycleofthesamelength π (k 1 1)=k 2 1, π (k 2 1)=k 3 1,..., π (k l 2 1)=k l 1 1, π (k l 1 1)=j 2 suchthatπ (j 2)=π(j 1) 1=k 1 1. Note that the structure of Lemma 6 completely corresponds to the procedure of creating the combinatorial minors. 4.Laplaceexpansionsoftype(15)ofper ω A, Cycl(A),S(A;n,k)andC(A;t 1,t 2,...,t n ) 1)per ω A.Considerallpermutationsπwiththeconditionπ(j)=1.Letj corresponds to j-th column of matrix A. Then the considered permutations correspondtodiagonalsofmatrixahavingthecommonposition(1,j).ifj=1, then,removingthefirstrowandcolumn,wediminishon1thenumberofcycles ofeverysuchpermutation,butalsowediminishon1thenumberofelements of permutations. Therefore, the decrement of permutations does not change. If j 2,considercontinuationofthesediagonalstothematrixofcombinatorial minora 1,j.Then,byLemma6,thenumberofcyclesofeveryitsdiagonaldoes

8 12 V. Shevelev not change and, consequently, the decrement is diminished by 1. This means that wehavethefollowingexpansionofper ω Aoverthefirstrow det ω A=a 1,1 (CM) 11 +ω a 1,j (CM) 1j, (20) where(cm) 1j, j 1,arecombinatorialminorsforper ω A. Note that, as for determinant(see Section 2), in order to receive the required enumerativeinformation,weshouldprohibittousetheidentitiesoftype1+ω+...+ω m 1 =0. 2)Cycl(A).Forn>1,hereweshouldignoreelementa 11.Considerallfull cyclesπwiththeconditionπ(j)=1.letjcorrespondstoj-thcolumnofmatrix A. Then the considered full cycles correspond to diagonals of matrix A having thecommonposition(1,j), j 2.Considercontinuationofthesediagonalsto thematrixofcombinatorialminora 1,j.Then,byLemma6,thenumberofcycles ofeveryitsdiagonaldoesnotchange,i.e.,theyarefullcyclesofofordern 1. Therefore,wehavethefollowingexpansionofCycle(A)overthefirstrowofA Cycle(A) = j=2 a 1,j (CM) 1j, (21) j=2 where(cm) 1j, j 1,arecombinatorialminorsforCycle(A). 3)S(A;n,k).Fromverycloseto1)arguments,wehavethefollowingexpansion ofs(a;n,k)overthefirstrowofa S(A;n,k)=a 1,1 (CM) (k 1) 11 + j=2 a 1,j (CM) (k) 1j, (22) where(cm) (k) 1j, j 1,arecombinatorialminorsofS(A;n,k). Notethatacloseto(22)formulawasfoundbytheauthorin[7]butusing much more complicated way. 4)C(t 1,t 2,...,t n ).Weneedlemma. Lemma7.Let beacycle.then {a 1j,a k11,a k2k 1,...,a kr,k r 1,a j,kr } (23) {a k11,a k2k 1,...,a kr,k r 1,a j,kr } (24)

9 Combinatorial minors for matrix functions and their applications 13 isacyclewithrespecttothemaindiagonalofthematrixofcombinatorialminora 1,j. Proof.AccordingtotheconstructionofA 1,j,themainitsdiagonalis {a 22,a 33,...,a j 1j 1,a j1,a j+1j+1,...,a nn }. (25) With respect to this diagonal we have the following contour which shows that(24) is, indeed, a cycle. {a k11 a j1 a jkr a krk r 1 a kr 1kr 2... a k2k 1 ( a k11)}. (26) Quiteanalogouslywecanprovethattoeveryanothercycleofadiagonalcontainingelementa 1j correspondthesamecyclewithrespecttothemaindiagonal ofthematrixa 1,j. LetAbe(0,1)squarematrixofordern.DenotebyC (r) (A;t 1,t 2,...,t n )apartialcycleindexofindex(9)ofpermutationsπ B(A)forwhich{1,π(1),π(2),..., π(r 1)}isacycleoflengthr.Thenwehave C (r) (A;t 1,t 2,...,t n )=C(A;t 1,t 2,...,t n ). (27) r=1 Therefore,itissufficienttogiveanexpansionofC (r) (A;t 1,t 2,...,t n ), r=1,...,n. Firstofall,notethat C (1) (A;t 1,t 2,...,t n )=a 11 t 1 C(A;t 1,t 2,...,t n 1 ). (28) Furthermore, using Lemmas 6-7, we have where C (r) (A;t 1,t 2,...,t n )= t r t r 1 a 1,j (CM) 1,j, (29) j=2 (CM) 1,j =C (r 1) (A 1,j;t 1,t 2,...,t n 1 ), j=2,...,n, (30) are the combinatorial minors for cycle index of permutations with restricted positions. Note that factor r t t r 1 in(29)correspondstothediminutionofthelength ofcycle(24)withrespecttolengthofcycle(23). Thus formulas(27)-(30) reduce the calculation of cycle index of n-permutationstothecalculationofcycleindexofn 1-permutationswitharathersimple computer realization of this procedure. Note that similar but much more complicated procedure was indicated by the author in[8].

10 14 V. Shevelev 5. An example of enumerating the permutations of classes 0,1,2 modulo 3 with restricted positions Let A= Consider class B(A) of permutations with restricted positions and find the distributionofthemoverclasses0,1,2modulo3.weuseω-permanentwithω=e 2πi 3 anditsexpansionoverelementsofthefirstrow,givenby(20).recallthat,for the receiving the required enumerative information, we should not use identities oftype1+ω+ω 2 =0. We have per ω A=per ω ( ω per ω per ω per ) ω = ( ) =per ω ω per ω per ω ω 2( ) per ω per ω ωper ω ω 2( ) per ω per ω ωper ω ω 2 per ω = 1 1 1

11 Combinatorial minors for matrix functions and their applications 15 =(ω+ω 2 )+(ω+ω 2 +1)+(2ω 2 +1)+ +(ω 2 +2+ω)+(2+2ω)+(ω 2 +1)+(ω 2 +1+ω)+(2+ω)+ +(ω+2ω 2 +1)+(ω 2 +2+ω)=13+9ω+10ω 2. ThusinB(A)wehave13permutationofclass0modulo3;9permutationsofclass 1modulo3and10permutationsofclass2modulo3. 6. On two sequences connected with Cycle(A) In summer of 2010, the author published two sequences A and A inoeis[10].a(n):=a179926(n)isdefinedasthenumberofpermutationsof allτ(n)divisorsofnoftheform:d 1 =n, d 2, d 3,..., d τ(n) suchthat di+1 d i is aprimeor1/primefori=1,...τ(n).notethata(n)isafunctionofexponents of prime power factorization of n only; moreover, it is invariant with respect to permutations of them. This sequence is equivalently defined as the number of ways, foragivenfinitemultisete,toget,beginningwithe,allsubmultisetsofe,if ineverystepweremoveorjoinoneelementofe.sequenceb(n):=a180026(n) differsfroma179926byanadditionalcondition: d τ(n) d 1 isaprime.intheequivalent formulationitcorrespondstotheconditionthatinthelaststepeisobtainedfrom a submultiset by joining one element. Note that, it is easy to prove that, knowing any permissible permutation of divisors,say,δ 1 =n, δ 2,..., δ τ(n) (suchthat δi+1 δ i isaprimeor1/prime),we can calculate b(n) using the following construction. Consider square(0,1) matrix B={b ij }oforderτ(n)inwhichb ij =1,if δi δ j isprimeor1/prime,andb ij =0, otherwise. Then b(n) = Cycle(B). In case of A179926, the construction is a little morecomplicated:a(n)=cycle(a),whereaisobtainedfrombbythereplacing thefirstitscolumnbythecolumnfrom1 s. NotealsothatA.Heinz[2]provedthat,inparticular,a(Π n i=1 p i),wherep i are distinct primes, equals to the number of Hamiltonian paths(or Gray codes) on n-cubewithamarkedstartingnode(seea003043in[10]),whileb(π n i=1 p i)equals to the number of directed Hamiltonian cycles on n-cube(see A in[10]). 7. Conclusive remarks It is interesting that together with the algorithm(27)-(30) for the evaluation ofcycleindexofpermutationswithrestrictedpositionsc(a;t 1,t 2,...,t n )thereis

12 16 V. Shevelev awaytofindanexplicitrepresentationforit.itappearsthatauniquewayfor that gives a so-called method of index of arrangements which was discovered by the author in[6]. This method was realized for finding explicit formulas for C(A;t 1,t 2,...,t n )(withconcreteexamples)in[9].actually,in[9]wegaveawide development of Riordan-Kaplansky theory of rook polynomials[5]. Acknowledgment The author is grateful to Alois Heinz for a computer realization of formula (21) and the calculations of large values of sequences A and A References 1. Abramowitz M., Stegun I.A.: Handbook on Special Functions. Dover Publ., New York HeinzA.,SeqfanDiscussionLists.Aug3and7, Kostrikin A.I.: Introduction to Algebra. Springer, New York Minc H.: Permanents. Addison-Wesley, Reading Riordan J.: An Introduction to Combinatorial Analysis, fourth printing. Wiley, New York ShevelevV.S.:Onamethodofconstructingrookpolynomialsandsomeofits applications. Combinatorial Analysis(MSU) 8(1989), (in Russian). 7. Shevelev V.S.: Enumerating the permutations with restricted positions and afixednumberofcycles.diskr.mat.,4,no.2(1992),3 22(inRussian). 8. Shevelev V.: Computing cyclic indicators of permutations with confined displacements. Dokl. Akad. Nauk of the Ukraine 7(1993), Shevelev V.: Development of the rook technique for calculating the cycle indices of(0,1)-matrices. Izvestia Vuzov of the North-Caucasus region, Nature Sciences(Mathematics) 4(1996), 21 28(in Russian). 10. Sloane N.J.A.: The On-Line Encyclopedia of Integer Sequences.