aa r X i v : . [ m a t h . C O ] S e p A Poset View of the Major Index
Richard Ehrenborg ∗ and Margaret Readdy † Abstract
We introduce the Major MacMahon map from Z h a , b i to Z [ q ], and show how this mapinteracts with the pyramid and bipyramid operators. When the Major MacMahon map is appliedto the ab -index of a simplicial poset, it yields the q -analogue of n ! times the h -polynomial ofthe poset. Applying the map to the Boolean algebra gives the distribution of the major indexon the symmetric group, a seminal result due to MacMahon. Similarly, when applied to thecross-polytope we obtain the distribution of one of the major indexes on signed permutationsdue to Reiner. Primary 06A07; Secondary 05A05, 52B05.
Key words and phrases.
The major index; permutations and signed permutations; the Booleanalgebra and the face lattice of a cross-polytope; simplicial posets; and principal specialization.
One hundred and one years ago in 1913 Major Percy Alexander MacMahon [9] (see also his collectedworks [11]) introduced the major index of a permutation π = π π · · · π n of the multiset M = { α , α , . . . , k α k } of size n to be the sum of the elements of its descent set, that is,maj( π ) = X π i >π i +1 i. He showed that the distribution of this permutation statistic is given by the q -analogue of themultinomial Gaussian coefficient, that is, the following identity holds: X π q maj( π ) = [ n ]![ α ]! · [ α ]! · · · [ α k ]! = (cid:20) nα (cid:21) , (1.1)where π ranges over all permutations of the multiset M and α is the composition ( α , α , . . . , α k ).Here [ n ]! = [ n ] · [ n − · · · [1] denotes the q -analogue of n !, where [ n ] = 1 + q + · · · + q n − .Many properties of the descent set of a permutation π , that is, Des( π ) = { i : π i > π i +1 } ,have been studied by encoding the set by its ab -word; see for instance [6, 12]. For a multiset ∗ Corresponding author: Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA, [email protected] , phone +1 (859) 257-4090, fax +1 (859) 257-4078. † Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA, [email protected] . π ∈ S M the ab -word is given by u ( π ) = u u · · · u n − , where u i = b if π i > π i +1 and u i = a otherwise.Inspired by this definition, we introduce the Major MacMahon map
Θ on the ring Z h a , b i of non-commutative polynomials in the variables a and b to the ring Z [ q ] of polynomials in thevariable q , by Θ( w ) = Y i : u i = b q i , for a monomial w = u u · · · u n and extend Θ to all of Z h a , b i by linearity. In short, the map Θsends each variable a to 1 and the variables b to q to the power of its position, read from left toright. A Swedish example is Θ( abba ) = q . Let P be a graded poset of rank n + 1 with minimal element b
0, maximal element b ρ . Let the rank difference be defined by ρ ( x, y ) = ρ ( y ) − ρ ( x ). The flag f -vector entry f S ,for S = { s < s < · · · < s k } a subset { , , . . . , n } , is the number of chains c = { b x < x Theorem 3.1. The Major MacMahon map Θ interacts with right multiplication by c , the deriva-tion G , the pyramid and the bipyramid operators as follows: Θ( w · c ) = (1 + q n +1 ) · Θ( w ) , (3.1)Θ( G ( w )) = q · [ n ] · Θ( w ) , (3.2)Θ(Pyr( w )) = [ n + 2] · Θ( w ) , (3.3)Θ(Bipyr( w )) = [2] · [ n + 1] · Θ( w ) , (3.4) where w is a homogeneous ab -polynomial of degree n . roof. It is enough to prove the four identities for an ab -monomial w of degree n . Directly we havethat Θ( w · a ) = Θ( w ) and Θ( w · b ) = q n +1 · Θ( w ). Adding these two identities yields equation (3.1).Assume that w consists of k b ’s. We label the n letters of w as follows: The k b ’s are labeled1 through k reading from right to left, whereas the n − k a ’s are labeled k + 1 through n readingleft to right. As an example, the word w = aababba is written as w w w w w w w .Identity (3.2) is a consequence of the following claim. Applying the derivation G only to theletter w i and then applying the Major MacMahon map yields q i · Θ( w ), that is,Θ( u · G ( w i ) · v ) = q i · Θ( u · w i · v ) , (3.5)where w is factored as u · w i · v . To see this, first consider when 1 ≤ i ≤ k . There are i b ’s to theright of w i including w i itself. They each are shifted one step to the right when replacing w i = b with G ( b ) = ab and hence we gain a factor of q i . The second case is when k + 1 ≤ i ≤ n . Then w i is an a and is replaced by ba under the derivation G . Assume that there are j b ’s to the rightof w i . When these j b ’s are shifted one step to the right they contribute a factor of q j . We alsocreate a new b . It has i − k − a ’s to the left and k − j b ’s to the left. Hence the position of thenew b is ( i − k − 1) + ( k − j ) + 1 = i − j and thus its contribution is q i − j . Again the factor is givenby q j · q i − j = q i , proving the claim. Now by summing over these n cases, identity (3.2) follows.Identity (3.3) is the sum of identities (3.1) and (3.2).To prove identity (3.4), we use a different labeling of the monomial w . This time label the k b ’swith the subscripts 0 through k − 1, rather than 1 through k . That is, in our example w = aababba is now labeled as w w w w w w w . We claim that for w = u · w i · v we have thatΘ( u · D ( w i ) · v ) = q i · [2] · Θ( w ) . The first case is 0 ≤ i ≤ k − 1. Then w i = b has i b ’s to its right. Thus when replacing b with ba there are i b ’s that are shifted one step, giving the factor q i . Similarly, when replacing w i with ab ,there are i + 1 b ’s that are shifted one step, giving the factor q i +1 . The sum of the two factors is q i · [2]. The second case is k + 1 ≤ i ≤ n . It is as the second case above when replacing w i with ba ,yielding the factor q i . When replacing w i with ab there is one more shift, giving q i +1 . Addingthese two subcases completes the proof of the claim.It is straightforward to observe thatΘ( c · w ) = q k · [2] · Θ( w ) . Calling this the case i = k , the identity (3.4) follows by summing the n + 1 cases 0 ≤ i ≤ n .Iterating equations (3.3) and (3.4) we obtain that the Major MacMahon map of the ab -indexof the n -dimensional simplex ∆ n and the n -dimensional cross-polytope C ∗ n . Corollary 3.2. The n -dimensional simplex ∆ n and the n -dimensional cross-polytope C ∗ n satisfy Θ(Ψ(∆ n )) = [ n + 1]! , Θ(Ψ( C ∗ n )) = [2] n · [ n ]! . Simplicial posets A graded poset P is simplicial if all of its lower order intervals are Boolean, that is, for all elements x < b b , x ] is isomorphic to the Boolean algebra B ρ ( x ) . It is well-known that all theflag information of a simplicial poset of rank n + 1 is contained in the f -vector ( f , f , . . . , f n ),where f = 1 and f i = f { i } for 1 ≤ i ≤ n . The h -vector, equivalently, the h -polynomial h ( P ) = h + h · q + · · · + h n · q n of a simplicial poset P , is defined by the polynomial relation h ( q ) = n X i =0 f i · q i · (1 − q ) n − i . See for instance [19, Section 8.3]. The h -polynomial and the bipyramid operation interact as follows: h (Bipyr( P )) = (1 + q ) · h ( P ) . We can now evaluate the Major MacMahon map on the ab -index of a simplicial poset. Theorem 4.1. For a simplicial poset P of rank n + 1 the following identity holds: Θ(Ψ( P )) = [ n ]! · h ( P ) . (4.1) Proof. Let B n ∪ { b } denote the Boolean algebra B n with a new maximal element added. Note that B n ∪ { b } is indeed a simplicial poset and its h -polynomial is 1. Furthermore, equation (4.1) holdsfor B n ∪ { b } sinceΘ(Ψ( B n ∪ { b } )) = Θ(Ψ( B n ) · a ) = Θ(Ψ( B n )) = [ n ]! = [ n ]! · h ( B n ∪ { b } ) . Also, if (4.1) holds for a poset P then it also holds for Bipyr( P ), since we haveΘ(Ψ(Bipyr( P ))) = [2] · [ n + 1] · Θ(Ψ( P )) = [2] · [ n + 1] · [ n ]! · h ( P ) = [ n + 1]! · h (Bipyr( P )) . Observe that both sides of (4.1) are linear in the h -polynomial. Hence to prove it for anysimplicial poset P it is enough to prove it for a basis of the span of all simplicial posets of rank n +1.Such a basis is given by the posets B n = n Bipyr i ( B n − i ∪ { b } ) o ≤ i ≤ n . This is a basis since the polynomials h (Bipyr i ( B n − i ∪ { b } )) = (1 + q ) i , for 0 ≤ i ≤ n , are a basisfor polynomials in the variable q of degree at most n .Finally, since every element in the basis is built up by iterating bipyramids of the posets B n ∪{ b } ,the theorem holds for all simplicial posets.Observe that the poset Bipyr i ( B n − i ∪ { b } ) is the face lattice of the simplicial complex consistingof the 2 i facets of the n -dimensional cross-polytope in the cone x , . . . , x n − i ≥ P , the h -vector is symmetric, that is, h i = h n − i . In other words,the h -polynomial is palindromic. Stanley [15] introduced the simplicial shelling components , that5s, the cd -polynomials ˇΦ n,i such that the cd -index of an Eulerian simplicial poset P of rank n + 1is given by Ψ( P ) = n X i =0 h i · ˇΦ n,i . (4.2)These cd -polynomials satisfy the recursion ˇΦ n, = Ψ( B n ) · c and ˇΦ n,i = G ( ˇΦ n − ,i − ); see [7,Section 8]. The Major MacMahon map of these polynomials is described by the next result. Corollary 4.2. The Major MacMahon map of the simplicial shelling components is given by Θ( ˇΦ n,i ) = q i · [2( n − i )] · [ n − . Proof. When i = 0 we have Θ( ˇΦ n, ) = Θ(Ψ( B n ) · c ) = (1 + q n ) · [ n ]! = [2 n ] · [ n − i ≥ n,i ) = Θ( G ( ˇΦ n − ,i − )) = q · [ n − · Θ( ˇΦ n − ,i − ) = q i · [2( n − i )] · [ n − Theorem 4.3. For an Eulerian poset P of rank n + 1 , the polynomial [2] ⌈ n/ ⌉ divides Θ(Ψ( P )) .Proof. It is enough to show this result for a cd -monomial w of degree n . A c in an odd position i of w yields a factor of 1 + q i . A d that covers an odd position i of w yields either q i − + q i or q i + q i +1 . Each of these polynomials contributes a factor of 1 + q . The result follows since thereare ⌈ n/ ⌉ odd positions. We now study how the Major MacMahon map behaves under the Cartesian product. Recall that fora graded poset P the ab -index Ψ( P ) encodes the flag f -vector information of the poset P . There isanother encoding of this information as a quasi-symmetric function. For further information aboutquasi-symmetric functions, see [17, Section 7.19].A composition α of n is a list of positive integers ( α , α , . . . , α k ) such that α + α + · · · + α k = n .Let Comp( n ) denote the set of compositions of n . There are three natural bijections between ab -monomials u of degree n , subsets S of the set { , , . . . , n } and compositions of n + 1. Given acomposition α ∈ Comp n +1 we have the subset S α , the ab -monomial u α and the ab -polynomial v α defined by S α = { α , α + α , . . . , α + · · · + α k − } ,u α = a α − · b · a α − · b · · · b · a α k − ,v α = ( a − b ) α − · b · ( a − b ) α − · b · · · b · ( a − b ) α k − . For S a subset of { , , . . . , n } let co( S ) denote associated composition.The monomial quasi-symmetric function M α is defined as the sum M α = X i
1. The image of γ is all quasi-symmetric functions without constantterm. Moreover, the image of the ab -monomial u α under γ is the fundamental quasi-symmetricfunction L α , that is, γ ( u α ) = L α . Another way to encode the flag vectors of a poset P is by the quasi-symmetric function of theposet. It is quickly defined as F ( P ) = γ (Ψ( P )). A more poset-oriented definition is the followinglimit of sums over multichains: F ( P ) = lim k −→∞ X b x ≤ x ≤···≤ x k = b t ρ ( x ,x )1 · t ρ ( x ,x )2 · · · t ρ ( x k − ,x k ) k . For more on the quasi-symmetric function of a poset, see [5].The stable principal specialization of a quasi-symmetric function is the substitution ps( f ) = f (1 , q, q , . . . ). Note that this is a homeomorphism, that is, ps( f · g ) = ps( f ) · ps( g ).For a composition α = ( α , α , . . . , α k ) let α ∗ denote the reverse composition, that is, α ∗ =( α k , . . . , α , α ). This involution extends to an anti-automorphism on QSym by M ∗ α M α ∗ .Define ps ∗ by the relation ps ∗ ( f ) = ps( f ∗ ). Informally speaking, this corresponds to the substitutionps ∗ ( f ) = f ( . . . , q , q, Theorem 5.1. For a homogeneous ab -polynomial w of degree n − the Major MacMahon map isgiven by Θ( w ) = (1 − q ) n · [ n ]! · ps ∗ ( γ ( w )) . (5.1) For a poset P of rank n this identity is Θ(Ψ( P )) = (1 − q ) n · [ n ]! · ps ∗ ( F ( P )) . (5.2) Proof. It is enough to prove identity (5.1) for an ab -monomial w of degree n − 1. Let α be thecomposition of n corresponding to the reverse monomial w ∗ . Furthermore, let e ( α ) be the sum P i ∈ S α ( n − i ). Note that e ( α ) is in fact the sum P i ∈ S i , where S is the subset associated withthe ab -monomial w . That is, we have q e ( α ) = Θ( w ). Equation (5.1) follows from Lemma 7.19.10in [17]. By applying the first identity to Ψ( P ), we obtain identity (5.2).Since the quasi-symmetric function is multiplicative under the Cartesian product, we have thenext result. 7 heorem 5.2. For two posets P and Q of ranks m , respectively n , the following identity holds: Θ(Ψ( P × Q )) = (cid:20) m + nn (cid:21) · Θ(Ψ( P )) · Θ(Ψ( Q )) . (5.3) Proof. The proof is a direct verification as follows:Θ(Ψ( P × Q )) = (1 − q ) m + n · [ m + n ]! · ps( F ( P ∗ × Q ∗ ))= (cid:20) m + nm (cid:21) · (1 − q ) m + n · [ m ]! · [ n ]! · ps( F ( P ∗ )) · ps( F ( Q ∗ ))= (cid:20) m + nm (cid:21) · Θ(Ψ( P )) · Θ(Ψ( Q )) . Define the quasi-symmetric function of type B ∗ of a graded poset P to be the expression F B ∗ ( P ) = X b ≤ x< b F ([ b , x ]) · s ρ ( x, b − . This is an element of the algebra QSym ⊗ Z [ s ] which we view as the quasi-symmetric functionsof type B ∗ . We view QSym B ∗ as a subalgebra of Z [ t , t , . . . ; s ], which is quasi-symmetric in thevariables t , t , . . . . For instance, a basis for QSym B ∗ is given by M α · s i where α ranges over allcompositions and i over all non-negative integers. Similar to the map γ : Z h a , b i −→ QSym, wedefine γ B ∗ : Z h a , b i −→ QSym B ∗ by γ B ∗ (cid:0) ( a − b ) α − · b · ( a − b ) α − · b · · · b · ( a − b ) α k − · b · ( a − b ) p (cid:1) = M α · s p , where α is the composition ( α , α , . . . , α k ). Similar to the relation γ (Ψ( P )) = F ( P ), we have γ B ∗ (Ψ( P )) = F B ∗ ( P ) . Furthermore, the type B ∗ quasi-symmetric function F B ∗ is multiplicative respect to the product ⋄ ∗ ,that is, F B ∗ ( P ⋄ ∗ Q ) = F B ∗ ( P ) · F B ∗ ( Q ); see [8, Theorem 13.3].Let f be a homogeneous quasi-symmetric function such that f · s j is a quasi-symmetric functionof type B ∗ . Define the stable principal specialization of the quasi-symmetric function f · s j of type B ∗ to be ps B ∗ ( f · s j ) = q deg( f ) · ps ∗ ( f ), where ps ∗ ( f ) = ps( f ∗ ). This is the substitution s = 1, t k = q , t k − = q , . . . as k tends to infinity, since f ( . . . , q , q , q ) = q deg( f ) · f ( . . . , q , q, P we have ps B ∗ ( F B ∗ ( P )) = X b ≤ x< b q ρ ( x ) · ps ∗ ( F ([ b , x ])) . (6.1)8 heorem 6.1. For a graded poset P of rank n + 1 the relationship between the Major MacMahonmap and the stable principal specialization of type B ∗ is given by Θ(Ψ( P )) = (1 − q ) n · [ n ]! · ps B ∗ ( F B ∗ ( P ∗ )) . (6.2) Especially, for a homogeneous ab -polynomial w of degree n the Major MacMahon map is given by Θ( w ) = (1 − q ) n · [ n ]! · ps B ∗ ( γ B ∗ ( w ∗ )) . (6.3) Proof. For the poset P we haveps ∗ ( F ( P )) = lim k →∞ X b x ≤ x ≤···≤ x k = b (cid:16) q k − (cid:17) ρ ( x ,x ) · · · (cid:0) q (cid:1) ρ ( x k − ,x k − ) · q ρ ( x k − ,x k − ) · ρ ( x k − ,x k ) = lim k →∞ X b x ≤ x ≤···≤ x k = b q ρ ( x k − ) · (cid:16) q k − (cid:17) ρ ( x ,x ) · · · q ρ ( x k − ,x k − ) · ρ ( x k − ,x k − ) = X b ≤ x ≤ b q ρ ( x ) · ps ∗ ( F ([ b , x ]))= X b ≤ x< b q ρ ( x ) · ps ∗ ( F ([ b , x ])) + q n +1 · ps ∗ ( F ( P )) . Rearranging terms yields X b ≤ x< b q ρ ( x ) · ps ∗ ( F ([ b , x ])) = (1 − q n +1 ) · ps ∗ ( F ( P ))= (1 − q n +1 ) · ps( F ( P ∗ ))= (1 − q n +1 ) · Θ(Ψ( P ))(1 − q ) n +1 · [ n + 1]!= Θ(Ψ( P ))(1 − q ) n · [ n ]! . Combining the last identity with (6.1) yields the desired result. Theorem 6.2. For two graded posets P and Q of ranks m + 1 , respectively n + 1 , the identityholds: Θ(Ψ( P ⋄ ∗ Q )) = (cid:20) m + nn (cid:21) · Θ(Ψ( P )) · Θ(Ψ( Q )) . (6.4) Proof. The proof is a direct verification as follows:Θ(Ψ( P ⋄ ∗ Q )) = (1 − q ) m + n · [ m + n ]! · ps B ∗ ( F B ∗ ( P ∗ ⋄ ∗ Q ∗ ))= (cid:20) m + nm (cid:21) · (1 − q ) m + n · [ m ]! · [ n ]! · ps B ∗ ( F B ∗ ( P ∗ )) · ps B ∗ ( F B ∗ ( Q ∗ ))= (cid:20) m + nm (cid:21) · Θ(Ψ( P )) · Θ(Ψ( Q )) . Permutations One connection between permutations and posets is via the concept of R -labelings. For moredetails, see [16, Section 3.14]. Let E ( P ) be the set of all cover relations of P , that is, E ( P ) = { ( x, y ) ∈ P : x ≺ y } . A graded poset P has an R -labeling if there is a map λ : E ( P ) −→ Λ, whereΛ is a linearly ordered set, such that in every interval [ x, y ] in P there is a unique maximal chain c = { x = x ≺ x ≺ · · · ≺ x k = y } such that λ ( x , x ) ≤ λ ( x , x ) ≤ · · · · · · ≤ λ ( x k − , x k ).For a maximal chain c in the poset P of rank n , let λ ( c ) denote the list ( λ ( x , x ), λ ( x , x ),. . . , λ ( x k − , x k )). The Jordan–H¨older set of P , denoted by J H ( P ), is the set of all the lists λ ( c )where c ranges over all maximal chains of P . The descent set of a list of labels λ ( c ) is the set ofpositions where there are descents in the list. Similarly, we define the descent word of λ ( c ) to be u λ ( c ) = u u · · · u n − where u i = b if λ ( x i − , x i ) > λ ( x i , x i +1 ) and u i = a otherwise.The bridge between posets and permutations is given by the next result. Theorem 7.1. For an R -labeling λ of a graded poset P we have that Ψ( P ) = X c u λ ( c ) , where the sum is over the Jordan–H¨older set J H ( P ) . This is a reformulation of a result of Bj¨orner and Stanley [3, Theorem 2.7]. The reformulationcan be found in [6, Lemma 3.1].As a corollary we obtain MacMahon’s classical result on the major index on a multiset; see [9].For a composition α of n let S α denote all the permutations of the multiset { α , α , . . . , k α k } . Corollary 7.2 (MacMahon) . For a composition α = ( α , α , . . . , α k ) of n the following identityholds: X π ∈ S α q maj( π ) = [ n ]![ α ]! · [ α ]! · · · [ α k ]! . Proof. Let P i denote the chain of rank α i for i = 1 , . . . , k . Furthermore, label all the cover relationsin P i with i . Let L denote the distributive lattice P × P × · · · × P k . Furthermore, let L inheritan R -labeling from its factors, that is, if x = ( x , x , . . . , x k ) ≺ ( y , y , . . . , y k ) = y let the label λ ( x, y ) be the unique coordinate i such that x i ≺ y i . Observe that the Jordan–H¨older set of L is S α . Direct computation yields Ψ( P i ) = a α i − , so the Major MacMahon map is Θ(Ψ( P i )) = 1.Iterating Theorem 5.2 evaluates the Major MacMahon map on L : X π ∈ S α q maj( π ) = Θ X π ∈ S α u ( π ) ! = Θ (Ψ( L )) = (cid:20) nα (cid:21) . For a vector r = ( r , r , . . . , r n ) of positive integers let an r -signed permutation be a list σ =( σ , σ , . . . , σ n +1 ) = (( j , π ) , ( j , π ) , . . . , ( j n , π n ) , 0) such that π π · · · π n is a permutation in thesymmetric group S n and the sign j i is from the set S π i = {− } ∪ { , . . . , r π i } . On the set of labels10 s s ss q q q ( − , i ) (2 , i ) ( r i , i )0 0 0 Figure 1: The poset P i with its R -labeling used in the proof of Corollary 7.3.Λ = { ( j, i ) : 1 ≤ i ≤ n, j ∈ S i } ∪ { } we use the lexicographic order with the extra condition that0 < ( j, i ) if and only if 0 < j . Denote the set of r -signed permutations by S r n . The descent set ofan r -signed permutation σ is the set Des( σ ) = { i : σ i > σ i +1 } and the major index is defined asmaj( σ ) = P i ∈ Des( σ ) i . Similar to Corollary 7.2, we have the following result. Corollary 7.3. The distribution of the major index for r -signed permutations is given by X σ ∈ S r n q maj( σ ) = [ n ]! · n Y i =1 (1 + ( r i − · q ) . Proof. The proof is the same as Corollary 7.2 except we replace the chains with the posets P i inFigure 1. Note that Ψ( P i ) = a + ( r i − · b . Let L be the lattice L = P ⋄ ∗ P ⋄ ∗ · · · ⋄ ∗ P n . Let L inherit the labels of the cover relations from its factors with the extra condition that the coverrelations attached to the maximal element receive the label 0. This is an R -labeling and the labelsof the maximal chains are exactly the r -signed permutations.For signed permutations, that is, r = (2 , , . . . , We suggest the following q, t -extension of the Major MacMahon map Θ. Define Θ q,t : Z h a , b i −→ Z [ q, t ] by Θ q,t ( w ) = Θ( w ) · w a =1 , b = t = Y i : u i = b q i · t, (8.1)for an ab -monomial w = u u · · · u n . Applying this map to the ab -index of the Boolean algebrayields one of the four types of q -Eulerian polynomials:Θ q,t (Ψ( B n )) = A maj , des n ( q, t ) = X π ∈ S n q maj( π ) t des( π ) . The following identity has been attributed to Carlitz [4], but goes back to MacMahon [10, Volume 2,Chapter IV, § X k ≥ [ k + 1] n · t k = A maj , des n ( q, t ) Q nj =0 (1 − t · q j ) . (8.2)11or recent work on the q -Eulerian polynomials, see Shareshian and Wachs [14]. It is natural to askif there is a poset approach to identity (8.2).In the second half of Section 7, before Corollary 7.3, we offer one way to define a major indexfor signed permutations. However, there are several different ways to extend the major index tosigned permutations. Two of our favorites are [1, 18]. Acknowledgements The authors thank the referee for his careful comments. The first author was partially supported byNational Security Agency grant H98230-13-1-0280. 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