An uppe bound on the numbe of high-dimensional pemutations Nathan Linial Zu Luia Abstact What is the highe-dimensional analog of a pemutation? If we think of a pemutation as given by a pemutation matix, then the following definition suggests itself: A d-dimensional pemutation of ode n is an n n n = [n] d+1 aay of zeos and ones in which evey line contains a unique 1 enty A line hee is a set of enties of the fom {(x 1,, x i 1, y, x i+1,, x d+1 ) n y 1} fo some index d + 1 i 1 and some choice of x j [n] fo all j i It is easy to obseve that a one-dimensional pemutation is simply a pemutation matix and that a two-dimensional pemutation is synonymous with an ode-n Latin squae We seek an estimate fo the numbe of d- dimensional pemutations Ou main esult is the following uppe bound on thei numbe ((1 + o(1)) n ) n d e d We tend to believe that this is actually the coect numbe, but the poblem of poving the complementay lowe bound emains open Ou main tool is an adaptation of Bègman s [1] poof of the Minc conjectue on pemanents Moe concetely, ou appoach is vey close in spiit to Schijve s [11] and Radhakishnan s [10] poofs of Bègman s theoem Depatment of Compute Science, Hebew Univesity, Jeusalem 91904, Isael e-mail: nati@cshujiacil Suppoted by ISF and BSF gants Depatment of Compute Science, Hebew Univesity, Jeusalem 91904, Isael e-mail: zluia@cshujiacil 1
1 Intoduction The pemanent of an n n matix A = (a ij ) is defined by P e(a) = σ S n n a i,σi Pemanents have attacted a lot of attention [9] They play an impotant ole in combinatoics Thus if A is a 0-1 matix, then P e(a) counts pefect matchings in the bipatite gaph whose adjacency matix is A They ae also of geat inteest fom the computational pespective It is #P -had to calculate the pemanent of a given 0-1 matix [12], and following a long line of eseach, an appoximation scheme was found [6] fo the pemanents of nonnegative matices Bounds on pemanents have also been studied at geat depth Van de Waeden conjectued that P e(a) n! fo evey n n n n doubly stochastic matix A, and this was established moe than fifty yeas late by Falikman and by Egoychev [4, 3] Moe ecently, Guvitz [5] discoveed a new conceptual poof fo this conjectue (see [8] fo a vey eadable pesentation) What is moe elevant fo us hee ae uppe bounds on pemanents These ae the subject of Minc s conjectue which was poved by Bègman Theoem 11 If A is an n n 0-1 matix with i ones in the i-th ow, then P e(a) n ( i!) 1/ i In the next section we eview Radhakishnan s poof, which uses the entopy method Ou plan is to imitate this poof fo a d-dimensional analogue of the pemanent To this end we need the notion of d-dimensional pemutations Definition 12 fom 1 Let A be an [n] d aay A line of A is vecto of the (A(i 1,, i j 1, t, i j+1,, i d )) n t=1, whee 1 j d and i 1,, i j 1, i j+1,, i d [n] 2
2 A d-dimensional pemutation (o d-pemutation) of ode n is an [n] d+1 aay P of zeos and ones such that evey line of P contains a single one and n 1 zeos Denote the set of all d-dimensional pemutations of ode n by S d,n Fo example, a two dimensional aay is a matix It has two kinds of lines, usually called ows and columns Thus a 1-pemutation is an n n 0-1 matix with a single one in each ow and a single one in each column, namely a pemutation matix A 2-pemutation is identical to a Latin squae and S 2,n is the same as the set L n, of ode-n Latin squaes We now explain the coespondence between the two sets If X is a 2-pemutation of ode n, then we associate with it a Latin squae L, whee L(i, j) as the (unique) index of a 1 enty in the line A(i, j, ) Fo moe on the subject of Latin squaes, see [13] The same definition yields a one-to-one coespondence between 3-dimensional pemutations and Latin cubes In geneal, d-dimensional pemutations ae synonymous with d-dimensional Latin hypecubes Fo moe on d-dimensional Latin hypecubes, see [14] To summaize, the following is an equivalent definition of a d-pemutation It is an [n] d aay with enties fom [n] in which evey line contains each i [n] exactly once We intechange feely between these two definitions accoding to context Ou main concen hee is to estimate S d,n, the numbe of d-pemutations of ode n By Stiling s fomula ( S 1,n = n! = (1 + o(1)) n ) n e As we saw, S 2,n is the numbe of ode n Latin squaes The best known estimate [13] is S 2,n = L n = ((1 + o(1)) n ) n 2 e 2 This elation is poved using bounds on pemanents Bégman s theoem fo the uppe bound, and the Falikman-Egoychev theoem fo the lowe bound This suggests Conjectue 13 S d,n = ((1 + o(1)) n ) n d e d In this pape we pove the uppe bound 3
Theoem 14 S d,n ((1 + o(1)) n ) n d e d As mentioned, ou method of poof is an adaptation of [10] We fist need Definition 15 M 2 if 1 An [n] d+1 0-1 aay M 1 is said to suppot an aay M 2 (i 1,, i d+1 ) = 1 M 1 (i 1,, i d+1 ) = 1 2 The d-pemanent of a [n] d+1 0-1 aay A is P e d (A) = The numbe of d-pemutations suppoted by A Note that in the one-dimensional case, this is indeed the usual definition of P e(a) It is not had to see that fo d = 1 the following theoem coincides with Bègman s theoem Theoem 16 Define the function f : N 0 N R ecusively by: f(0, ) = log(), whee the logaithm is in base e f(d, ) = 1 f(d 1, k) Let A be an [n] d+1 0-1 aay with i1,i d ones in the line A(i 1,, i d, ) Then P e d (A) ) i 1,,i d e f(d,i1,id We will deive below faily tight bounds on the function f that appeas in theoem 16 It is then an easy matte to pove theoem 14 by applying theoem 16 to the all-ones aay What about poving a matching lowe bound on S d,n (and thus poving conjectue 13)? In ode to follow the footsteps of [13], we would need a lowe bound on P e d (A), namely, a highe-dimensional analog of the van de Waeden conjectue The enties of a multi-stochastic aay ae nonnegative eals and the sum of enties along evey line is 1 This is the highe-dimensional countepat of a doubly-stochastic matix It should be clea how to extend the notion of P e d (A) to eal-valued aays In this appoach we would need a lowe bound on P e d (A) that holds fo evey multi-stochastic aay A 4
Howeve, this attempt (o at least its most simplistic vesion) is bound to fail An easy consequence of Hall s theoem says that a 0-1 matix in which evey line o column contains the same (positive) numbe of 1-enties, has a positive pemanent (We still do not know exactly how small such a pemanent can be, see [8] fo moe on this) Howeve, the highe dimensional analog of this is simply incoect Thee exist multi-stochastic aays whose d-pemanent vanishes, as can easily be deduced eg, fom [7] We can, howeve, deive a lowe bound of S d,n exp(ω(n d )) fo even n Conside the following constuction: Let n be an even intege, and let P be a d-pemutation of ode [ ] n d 2 It is easy to see that such a P exists Simply set P (i 1,, i d ) = (i 1 + + i d ) mod n 2 Now we constuct a d-pemutation Q of ode [n] d by eplacing each element of P with a [2] d block If P (i 1,, i d ) = j, then the coesponding block contains the values j and j + n It is easy to see that thee ae exactly 2 two ways to aange these values in each block, and that Q is indeed a d- pemutation of ode [n] d Thee ae ( ) n d 2 blocks, and so the numbe of possible Q s is 2 ( n 2 ) d Fo a constant d this is exp(ω(n d )) In section 2 we pesent Radhakishnan s poof of the Bègman bound In section 3 we pove theoem 16 In section 4 we use this bound to pove theoem 14 2 Radhakishnan s poof of Bègman s theoem 21 Entopy - Some basics We eview the basic mateial concening entopy that is used hee and efe the eade to [2] fo futhe infomation on the topic Definition 21 The entopy of a discete andom vaiable X is given by H(X) = ( ) 1 P(X = x) log P(X = x) x 5
Fo andom vaiables X and Y, the conditional entopy of X given Y is H(X Y ) = E[H(X Y = y)] = y P(Y = y)h(x Y = y) In this pape we will always conside the base e entopy of X which simply means that the logaithm is in base e Theoem 22 1 If X is a discete andom vaiable, then H(X) log ange(x), with equality iff X has a unifom distibution 2 If X 1,, X n is a sequence of andom vaiables, then H(X 1,, X n ) = H(X i X 1,, X i 1 ) 3 The inequality H(X Y ) H(X f(y )) holds fo evey two discete andom vaiables X and Y and evey eal function f( ) The following is a geneal appoach using entopy that is useful fo a vaiety of appoximate counting poblems Suppose that we need to estimate the cadinality of some set S If X is a andom vaiable which takes values in S unde the unifom distibution on S, then H(X) = log( S ) So, a good estimate on H(X) yields bounds on S This appoach is the main idea of both Radhakishnan s poof and ou wok 22 Radhakishnan s poof Let A be an n n 0-1 matix with i ones in the i-th ow Ou aim is to pove the uppe bound n P e(a) ( i!) 1 i 6
Let M be the set of pemutation matices suppoted by A, and let X be a unifomly sampled andom element of M Ou plan is to evaluate H(X) using the chain ule and estimate M using the fact (theoem 22) that H(X) = log( M ) Let X i be the unique index j such that X(i, j) = 1 We conside a pocess whee we scan the ows of X in sequence and estimate H(X) = H(X 1,, X n ) using the chain ule in the coesponding ode To cay out this plan, we need to bound the contibution of the tem involving X i conditioned on the peviously obseved ows That is, we wite H(X) = H(X i X 1,, X i 1 ) Let Let R i be the set of indices of the 1-enties in A s i-th ow That is, R i = {j : A(i, j) = 1} Z i = {j R i : X i = j fo some i < i} Note that X i R i, because X is suppoted by A In addition, given that we have aleady exposed the values X i fo i < i, it is impossible fo X i to take any value j Z i, o else the column X(, j) contains moe than a single 1-enty Theefoe, given the vaiables that pecede it, X i must take a value in R i Z i The cadinality N i = R i Z i is a function of X 1,, X i 1 and so by theoem 22, = H(X) = H(X i X 1,, X i 1 ) P(X 1 = x 1,, X i 1 = x i 1 )H(X i X 1 = x 1,, X i 1 = x i 1 ) x 1,,x i 1 P(X 1 = x 1,, X i 1 = x i 1 ) log(n i ) x 1,,x i 1 = E X1,,X i 1 [log(n i )] = E X [log(n i )] 7
It is not clea how we should poceed fom hee, fo how can we bound log(n i ) fo a geneal matix? Moeove, diffeent odeings of the ows will give diffeent bounds We use this fact to ou advantage and conside the expectation of this bound ove all possible odeings Associated with a pemutation σ S n is an odeing of the ows whee X j is evealed befoe X i if σ(j) < σ(i) We edefine Z i and N i to take the odeing σ into account Let Z i (σ) = {j R i : X i = j fo some σ(i ) < σ(i)} N i (σ) = R i Z i (σ) Then N i (σ) is the numbe of available values fo X i, given all the vaiables X j fo j such that σ(j) < σ(i) As befoe, using the chain ule we obtain the inequality H(X) = H(X i X j : σ(j) < σ(i)) E X [log(n i (σ))] The inequality emains tue if we take the expected value of both sides when σ is a andom pemutation sampled fom the unifom distibution on S n H(X) E σ [E X [log(n i (σ))]] = E X [E σ [log(n i (σ))]] Thus, the bound we get on H(X) depends on the distibution of the andom vaiable N i (σ) The final obsevation that we need is that the distibution of N i (σ) is vey simple and that it does not depend on X Consequently we can eliminate the step of taking expectation with espect to the choice of X Let us fix a specific X Let W i denote the set of i 1 ow indices j i fo which X j R i Note that N i is equal to i minus the numbe of indices in W i that pecede i in the andom odeing σ Since σ was chosen unifomly, this numbe is distibuted unifomly in {0,, i 1} Thus, N i is unifom on the set {1,, i } Theefoe Hence E σ [log(n i (σ))] = H(X) i [ ] 1 E X log( i!) = i 1 i log(k) = 1 i log( i!) 8 1 i log( i!)
which implies the Bègman bound 3 The d-dimensional case 31 An infomal discussion The coe of the above-descibed poof of the Bègman bound can be viewed as follows Let us pick fist a 1-pemutation X that is contained in the matix A and conside the set R i of the i 1-enties in A s i-th ow Thee ae exactly i indices j fo which X j R i The andom odeing of the ows detemines which of these will pecede the i-th ow (o will cast its shadow on the i-th ow) The andom numbe u i of ows that cast a shadow on the i-th ow is unifomly distibuted in the ange {0,, i 1} The contibution of this ow to the uppe bound on H(X) is E σ [log N i ], whee N i = i u i is the numbe of 1-enties in the i-th ow that ae still unshaded The expectation of log N i is exactly 1 i i j=1 log j = 1 i log( i!) How should we modify this agument to deal with d-dimensional pemutations? We fix a d-pemutation X that is contained in A and conside a andom odeing of all lines of the fom A(i 1,, i d, ) Given such an odeing, we use the chain ule to deive an uppe bound on H(X) Each odeing yields a diffeent bound Howeve, as in the one dimensional case, the key insight is that aveaging ove all possible odeings (in a class that we late define) gives us a simple bound on H(X) The oveall stuctue of the agument emains the same We conside a concete line A(i 1,, i d, ) Its contibution to the estimate of the entopy is log N whee N is the numbe of 1-enties that emain unshaded at the time (accoding to the chosen odeing) at which we compute the coesponding tem in the chain ule fo the entopy Howeve, now shade can fall fom d diffeent diections The contibution of the line to the entopy will be the expected logaithm of the numbe of ones that emain unshaded afte each of the d dimensions has cast its shade on it The lines ae odeed by a andom lexicogaphic odeing At the coasest level lines ae odeed accoding to thei fist coodinate i 1 This odeing is chosen unifomly fom S n To undestand how many 1 s emain unshaded in a given line, we fist conside the shade along the fist coodinate If it initially has 1-enties, then the numbe of unshaded 1-enties afte this stage is unifomly distibuted on [] We then ecuse with the emaining 9
1-enties and poceed on the subcube of codimension 1 that is defined by the value of the fist coodinate It is not had to see how the ecusive expession fo f(d, ) eflects this calculation 32 In detail Let A be a [n] d+1 -dimensional aay of zeos and ones, and X is a andom d-pemutation sampled unifomly fom the set of d-pemutations contained in A Then H(X) = log(p e d (A)) by theoem 22 and again we seek an uppe bound on H(X) We think of X as an [n] d aay each line of which contains each membe of [n] exactly once The poof does its accounting using lines of the fom A(i 1,, i d, ), ie, lines in which the (d + 1)-st coodinate vaies Such a line is specified by i = (i 1,, i d ) The andom vaiable X i is defined to be the value of X(i 1,, i d ) We think of the vaiables X i as being evealed to us one by one Thus, X i1,,i d must belong to R i = R i1,,i d = {j : A(i 1,, i d, j) = 1} the set of 1-enties in this line In the poof we scan these lines in a paticula andomly chosen ode Let us ignoe this issue fo a moment and conside some fixed odeing of these lines Initially, the numbe of 1-enties in this line is i As we poceed, some of these 1 s become unavailable to X i, since choosing them would esult in a conflict with the choice made in some peviously evealed line We say that these 1 s ae in the shade of peviously consideed lines This shade can come fom any of the d possible diections Thus we denote by Z i R i the set of the indices of the 1-enties in R i that ae unavailable to X i given the values of the peceding vaiables We can expess Z i = d Zk i whee enties in Zi k ae shaded fom diection k Namely, a membe j of R i belongs to Zi k if thee is an aleady scanned line indexed by i with X i = j and whee i and i coincide on all coodinates except the k-th Thus, given the values of the peviously consideed vaiables, thee ae at most N i = R i Z i values that ae available to X i We next tun to the andom odeing of the lines Now, howeve, we do not select a completely andom odeing, but opt fo a andom lexicogaphic 10
odeing Namely, we select d andom pemutations σ 1,, σ d S n The line A(i 1,, i d, ) pecedes A(i 1,, i d, ) if thee is a k [n] such that σ k(i k ) < σ k (i k ) and i j = i j fo all j < k Thus a choice of the odeings σ k induces a total ode on the lines A(i 1,, i d, ) Denote this ode by That is, we wite i j if i comes befoe j We wite i k j if i j and i and j diffe only in the k-th coodinate We think of X i as being evealed to us accoding to this ode We tun to the definition of R i, Zi k and N i Thei definitions ae affected by the chosen odeing of the lines In addition, fo easons to be made clea late, we genealize the definition of N i It is defined as the numbe of values available to X i (given the peceding lines) fom a given index set W R i In the discussion below, we fix X, a d-pemutation that is contained in A Definition 31 The index set of the 1-enties in the line A(i 1,, i d, ) is denoted by R i = R i1,,i d = {j : A(i 1,, i d, j) = 1}, and its cadinality is i = R i Let W R i with i = (i 1,, i d ), and suppose that X i W Fo a given odeing, let Z k i (X, ) = {j R i : X i = j fo some i k i} N i (W, X, ) = W d Z k i (X, ) Thus, N i is a function of W R i, X and the odeing Each vaiable X i specifies a 1 enty of the line A(i 1,, i d, ) The enty thus specified must confom to the values taken by the peceding vaiables Namely, no line of X can contain moe than a single 1 enty We conside the numbe of values that the vaiable X i can take, given the values that pecede it Fix an index tuple i = (i 1,, i d ) The vaiable X i must specify an index i d+1 with A(i 1,, i d+1 ) = 1, ie, an element of R i Conside some element j R i If X i = j, fo some i k i and k d then clealy X i j, o else the line X(i 1,, i k 1,, i k+1,, i d ) contains moe than a single j-enty In othe wods, X i cannot specify an element of Zi k (X, ) and is esticted to the set R i d Zk i (X, ) Theefoe, thee ae at most N i(r i, X, ) possible values that X i can take given the vaiables that pecede it in the ode 11
Fo a given ode, we can use the chain ule to deive H(X) = i H(X i X j : j i) By theoem 22, H(X i X j : j i) = E Xj :j i [H(X i X j = x j : j i)] E Xj :j i [log(n i (R i, X, ))] = E X [log(n i (R i, X, ))] The last equality holds because N i depends only on the lines of X that pecede X i, and so taking the expectation ove the est of X doesn t change anything As in the one dimensional case, the next step is to take the expectation of both sides of the above inequality ove H(X) i E [E X [log(n i (R i, X, ))]] = i E X [E [log(n i (R i, X, ))]] The key to unaveling this expession is the insight that the andom vaiable N i has a simple distibution (as a function of ), and moeove, that this distibution does not depend on X Recall that in the one dimensional case, we obtained the distibution of N i as follows Initially, the numbe of ones in the i-th ow was i Then the ows peceding the i-th ow wee evealed, and some of the ones in the i-th ow became unavailable to X, because some othe ow had placed a one in thei column We defined N i = R i Z i (σ) The size of Z i (σ) was shown to be unifomly distibuted ove {0,, i 1}, and thus the distibution of N i was shown to be unifom ove {1,, i } A simila agument woks in the d dimensional case, but the distibution of N i is no longe unifom Recall that the function f is defined ecusively by f(0, ) = log() f(d, ) = 1 f(d 1, k) 12
Claim 32 Let X be a d-pemutation, i = (i 1,, i d ) and let W R i be an index set such that X i W Then E [log(n i (W, X, ))] depends only on d and = W, and E [log(n i (W, X, ))] = f(d, ) Poof The poof poceeds by induction on d Fist, note that if W = and d = 0, then N i (W, X, ) = W = by definition, and theefoe E [log(n i (W, X, ))] = log() = f(0, ) In ode to poceed with the induction step, we must descibe N i (W, X, ) in tems of paametes of dimension d 1 instead of d To this end we need the following definitions: X = X(i 1,,, ) Note that X is a (d 1)-dimensional pemutation W = W Z 1 i (X, ) Note that W actually depends only on σ 1, the odeing of the fist coodinate Let i = (i 1,, i d 1 ) = (i 2,, i d ) Given an odeing, let be the odeing on the index tuples (i 1,, i d 1 ) defined by the odeings σ 2, σ 3,, σ d Note that fo evey X,W, i and we have N i (W, X, ) = N i (W, X, ) This equality follows diectly fom the definition of N Now, E [log(n i (W, X, ))] = E σ1 [E [log(n i (W, X, ))]] = E σ1 [E [log(n i (W, X, ))]] = E σ1 [f(d 1, W )] The last step follows fom the induction hypothesis Consequently, E [log(n i (W, X, ))] = k P( W = k)f(d 1, k) The only emaining question is to detemine the distibution of W as a function of σ 1 Note, howeve, that we have aleady answeed this question in the one dimensional poof, namely, W is unifomly distibuted on {1,, } Indeed, W = W Zi 1(X, ), and Z1 i (X, ) is the set of indices s such that: 13
Fo some j W, X(s, i 2,, i d ) = j (thee ae 1 such indices, one fo each j W ) The andom odeing σ 1 places s befoe i 1 In a andom odeing, the position of i 1 is unifomly distibuted Theefoe Zi 1(X, ) is unifomly distibuted on {0,, 1}, and P( W = k) = 1 fo evey 1 k Putting this togethe, we have shown that E [log(n i (W, X, ))] = 1 f(d 1, k) = f(d, ) In conclusion, we have shown that H(X) i E X [E [log(n i (R i, X, ))]] = i E X [f(d, i )] = i f(d, i ), is the numbe of ones in the vecto A(i 1,, i d, ) Thee- whee i = i1,,i d foe, P e d (A) i e f(d, i) 4 The numbe of d-pemutations An uppe bound As mentioned, the uppe bound on the numbe of d-dimensional pemutations is deived by applying theoem 16 to the all-ones aay J The main technical step is a deivation of an uppe bound on the function f(d, ) Theoem 41 Fo evey d thee exist constants c d and d such that fo all d, log d () f(d, ) log() d + c d One possible choice that we adopt hee is d = e d fo evey d, c 1 = 5, c 2 = 8, and c d = d3 (11) d fo d 3 d! 14
Poof A staightfowad induction on d yields the weake bound f(d, ) log() fo all d, Fo d = 0 thee is equality and the geneal case follows since f(d, ) = 1 f(d 1, k) 1 log(k) log() This simple bound seves us to deal with the ange of small s (below d 1 ) We tun to the main pat of the poof f(d, ) = 1 f(d 1, k) = 1 d 1 f(d 1, k) + k= d 1 +1 f(d 1, k) 1 log d 1 (k) d 1 log( d 1 ) + log(k) (d 1) + c d 1 k k= d 1 [ ] 1 log d 1 (k) d 1 log( d 1 ) + d 1 (d 1) + log(k) (d 1) + c d 1 k ξ + 1 log(!) (d 1) + c d 1 log d 1 (k) k whee ξ = d 1 log( d 1 )+ d 1 (d 1) = 2(d 1)e d 1 It is easily veified that fo d 3 thee holds log(!) log() + 2 log() We can poceed with ξ + log() + 2 log() d + c d 1 log d 1 (k) k log d 1 (x)dx x = We now bound the sum log d 1 (k) by means of the integal k 1 Note that the integand is unimodal and its maximal value is γ = d log d () ( d 1 e ) d 1 Thus, c d 1 log d 1 (k) k c d 1 ( log d () d ) + γ Putting this togethe, we have the inequality ( ) 2 log() + ξ + c d 1 γ + logd () d f(d, ) log() d + 15
Theefoe it is sufficient to choose c d such that fo evey e d ( ) 2 log() + ξ + c d 1 γ + logd () c d log d () d ie, ( 2 log d 1 () + ξ γ log d () + c d 1 log d () + 1 ) c d d The left hand side of the above inequality is clealy a deceasing function of Theefoe it is sufficient to veify the inequality fo = e d Plugging this and the values of the constants ξ and γ into the left hand side of the above inequality, we get Thus, we may take 2 2(d 1)ed 1 + dd 1 d d ( 1 + 1 e d 1 d c d = ) cd 1 ( (d 1) d 1 + c d 1 + 1 ) e d 1 d d d ( 2 ( e ) ) d + d d + d d ( 1 + 1 ) ( cd 1 2 ( e ) ) d e d 1 d + d d + d d Calculating c d using this ecusion and the fact that c 0 = 0, we get that c 1 = 2 + e 5, c 2 8, and c d d3 (11) d fo 3 d 10 Poceeding by d! induction, ( c d = 1 + 1 ) ( (d 1) 3 (11) d 1 2 ( e ) ) d + d e d 1 d! d + d d (11)d (d 1) 3 ( e d (11) + 2d d! d) d (d 1) 3 + 2d 2 (11)d d 3 d! d! In the inequality befoe the last one, we used the fact that fo d 10, ( e ) d d d d! Fo the [n] d+1 all ones aay J, i1,,i d = n fo evey tuple (i 1,, i d ), and so fo lage enough n we have the bound P e d (J) i 1,,i d e f(d,n) = ( e f(d,n) ) nd 16 ( [ log d ]) n d (n) exp log(n) d + c d n
log Fo a constant d, letting n go to infinity, c d (n) d = o(1) and theefoe the n numbe of d-pemutations is at most ((1 + o(1)) n ) n d e d Refeences [1] L M Bègman, Cetain popeties of nonnegative matices and thei pemanents, Dokl Akad Nauk SSSR 211 (1973), 27-30 MR MR0327788 (48 #6130) [2] T M Cove and J A Thomas, Elements of Infomation Theoy, Wiley, New Yok, 1991 [3] GP Egoichev, Poof of the van de Waeden conjectue fo pemanents, Sibeian Math J 22 (1981), 854-859 [4] DI Falikman, A poof of the van de Waeden conjectue egading the pemanent of a doubly stochastic matix, Math Notes Acad Sci USSR 29 (1981), 475-479 [5] L Guvits, Van de Waeden/SchijveValiant like conjectues and stable (aka hypebolic) homogeneous polynomials: one theoem fo all With a coigendum, Electon J Combin 15 (2008), R66 (26 pp) [6] M Jeum, A Sinclai, and E Vigoda, A polynomial-time appoximation algoithm fo the pemanent of a matix with nonnegative enties, J ACM, 671-697 [7] M Kochol, Relatively naow Latin paallelepipeds that cannot be extended to a Latin cube, As Combin 40 (1995), 247260 [8] Lauent, M and Schijve, A, On Leonid Guvitss poof fo pemanents, The Ameican Mathematical Monthly 10 (2010), 903 911 [9] H Minc, Pemanents, Encyclopedia of Mathematics and Its Applications Vol 6, Addison-Wesley, Reading, Mass, 1978 [10] Jaikuma Radhakishnan, An entopy poof of Begman s theoem, J Combinatoial Theoy Se A 77 (1997), no 1, 80-83 MR MR1426744 (97m:15006) 17
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