Enumeration of some particular quintuple persymmetric matrices over F_2 by rank
aa r X i v : . [ m a t h . N T ] S e p ENUMERATION OF SOME PARTICULAR QUINTUPLEPERSYMMETRIC MATRICES OVER F BY RANK
JORGEN CHERLY
R´esum´e.
Dans cet article nous comptons le nombre de certaines quin-tuples matrices persym´etriques de rang i sur F . Abstract.
In this paper we count the number of some particular quin-tuple persymmetric rank i matrices over F . Contents
1. Introduction 32. Notation and Preliminaries 32.1. Some notations concerning the field of Laurent Series F (( T − )) 32.2. Some results concerning n-times persymmetric matrices over F NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK3 Introduction
In this paper we propose to compute in the most simple case the numberof quintuple persymmetric matrices with entries in F of rank iThat is to compute the number Γ × ki of quintuple persymmetric matri-ces in F of rank i (0 i inf(10 , k )) of the below form.(1.1) α (1)1 α (1)2 α (1)3 α (1)4 α (1)5 . . . α (1) k α (1)2 α (1)3 α (1)4 α (1)5 α (1)6 . . . α (1) k +1 α (2)1 α (2)2 α (2)3 α (2)4 α (2)5 . . . α (2) k α (2)2 α (2)3 α (2)4 α (2)5 α (2)6 . . . α (2) k +1 α (3)1 α (3)2 α (3)3 α (3)4 α (3)5 . . . α (3) k α (3)2 α (3)3 α (3)4 α (3)5 α (3)6 . . . α (3) k +1 α (4)1 α (4)2 α (4)3 α (4)4 α (4)5 . . . α (4) k α (4)2 α (4)3 α (4)4 α (4)5 α (4)6 . . . α (4) k +1 α (5)1 α (5)2 α (5)3 α (5)4 α (5)5 . . . α (5) k α (5)2 α (5)3 α (5)4 α (5)5 α (5)6 . . . α (5) k +1 We remark that this paper is based on the results in the author’s paper [12]2.
Notation and Preliminaries
Some notations concerning the field of Laurent Series F (( T − )) . We denote by F (cid:0)(cid:0) T − (cid:1)(cid:1) = K the completion of the field F ( T ) , the field ofrational fonctions over the finite field F , for the infinity valuation v = v ∞ defined by v (cid:0) AB (cid:1) = degB − degA for each pair (A,B) of non-zero poly-nomials. Then every element non-zero t in F (cid:0)(cid:0) T (cid:1)(cid:1) can be expanded in aunique way in a convergent Laurent series t = P − v ( t ) j = −∞ t j T j where t j ∈ F . We associate to the infinity valuation v = v ∞ the absolute value | · | ∞ defined by | t | ∞ = | t | = 2 − v ( t ) . We denote E the Character of the additive locally compact group F (cid:0)(cid:0) T (cid:1)(cid:1) defined by E (cid:0) − v ( t ) X j = −∞ t j T j (cid:1) = ( t − = 0 , − t − = 1 . We denote P the valuation ideal in K , also denoted the unit interval of K , i.e. the open ball of radius 1 about 0 or, alternatively, the set of all Laurent JORGEN CHERLY series X i ≥ α i T − i ( α i ∈ F )and, for every rational integer j, we denote by P j the ideal { t ∈ K | v ( t ) > j } . The sets P j are compact subgroups of the additive locally compact group K . All t ∈ F (cid:16)(cid:16) T (cid:17)(cid:17) may be written in a unique way as t = [ t ] + { t } , [ t ] ∈ F [ T ] , { t } ∈ P (= P ) . We denote by dt the Haar measure on K chosen so that Z P dt = 1 .Let ( t , t , . . . , t n ) = (cid:0) − ν ( t ) X j = −∞ α (1) j T j , − ν ( t ) X j = −∞ α (2) j T j , . . . , − ν ( t n ) X j = −∞ α ( n ) j T j (cid:1) ∈ K n . We denote ψ the Character on ( K n , +) defined by ψ (cid:0) − ν ( t ) X j = −∞ α (1) j T j , − ν ( t ) X j = −∞ α (2) j T j , . . . , − ν ( t n ) X j = −∞ α ( n ) j T j (cid:1) = E (cid:0) − ν ( t ) X j = −∞ α (1) j T j (cid:1) · E (cid:0) − ν ( t ) X j = −∞ α (2) j T j (cid:1) · · · E (cid:0) − ν ( t n ) X j = −∞ α ( n ) j T j (cid:1) = ( α (1) − + α (2) − + . . . + α ( n ) − = 0 − α (1) − + α (2) − + . . . + α ( n ) − = 12.2. Some results concerning n-times persymmetric matrices over F . Set ( t , t , . . . , t n ) = (cid:0) X i ≥ α (1) i T − i , X i ≥ α (2) i T − i , X i ≥ α (3) i T − i , . . . , X i ≥ α ( n ) i T − i (cid:1) ∈ P n . Denote by D ... × k ( t , t , . . . , t n )the following 2 n × k n-times persymmetric matrix over the finite field F NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK5 (2.1) α (1)1 α (1)2 α (1)3 α (1)4 α (1)5 α (1)6 . . . α (1) k α (1)2 α (1)3 α (1)4 α (1)5 α (1)6 α (1)7 . . . α (1) k +1 α (2)1 α (2)2 α (2)3 α (2)4 α (2)5 α (2)6 . . . α (2) k α (2)2 α (2)3 α (2)4 α (2)5 α (2)6 α (2)7 . . . α (2) k +1 α (3)1 α (3)2 α (3)3 α (3)4 α (3)5 α (3)6 . . . α (3) k α (3)2 α (3)3 α (3)4 α (3)5 α (3)6 α (3)7 . . . α (3) k +1 ... ... ... ... ... ... ... ... α ( n )1 α ( n )2 α ( n )3 α ( n )4 α ( n )5 α ( n )6 . . . α ( n ) k α ( n )2 α ( n )3 α ( n )4 α ( n )5 α ( n )6 α ( n )7 . . . α ( n ) k +1 We denote by Γ ... × ki the number of rank i n-times persymmetric matricesover F of the above form :Let f ( t , t , . . . , t n ) be the exponential sum in P n defined by( t , t , . . . , t n ) ∈ P n −→ X degY ≤ k − X degU ≤ E ( t Y U ) X degU ≤ E ( t Y U ) . . . X degU n ≤ E ( t n Y U n ) . Then f k ( t , t , . . . , t n ) = 2 n + k − rank (cid:2) D ... × k ( t ,t ,...,t n ) (cid:3) Hence the number denoted by R ( k ) q,n of solutions( Y , U (1)1 , U (1)2 , . . . , U (1) n , Y , U (2)1 , U (2)2 , . . . , U (2) n , . . . Y q , U ( q )1 , U ( q )2 , . . . , U ( q ) n ) ∈ ( F [ T ]) ( n +1) q of the polynomial equations Y U (1)1 + Y U (2)1 + . . . + Y q U ( q )1 = 0 Y U (1)2 + Y U (2)2 + . . . + Y q U ( q )2 = 0... Y U (1) n + Y U (2) n + . . . + Y q U ( q ) n = 0 JORGEN CHERLY ⇔ U (1)1 U (2)1 . . . U ( q )1 U (1)2 U (2)2 . . . U ( q )2 ... ... ... ... U (1) n U (2) n . . . U ( q ) n Y Y ... Y q = satisfying the degree conditions degY i ≤ k − , degU ( i ) j ≤ , f or ≤ j ≤ n ≤ i ≤ q is equal to the following integral over the unit interval in K n Z P n f qk ( t , t , . . . , t n ) dt dt . . . dt n . Observing that f ( t , t , . . . , t n ) is constant on cosets of Q nj =1 P k +1 in P n theabove integral is equal to(2.2) 2 q (2 n + k ) − ( k +1) n inf(2 n,k ) X i =0 Γ ... × ki − iq = R ( k ) q,n Recall that R ( k ) q,n is equal to the number of solutions of the polynomial system U (1)1 U (2)1 . . . U ( q )1 U (1)2 U (2)2 . . . U ( q )2 ... ... ... ... U (1) n U (2) n . . . U ( q ) n Y Y ... Y q = (2.3) satisfying the degree conditions degY i ≤ k − , degU ( i ) j ≤ , f or ≤ j ≤ n ≤ i ≤ q From (2.2) we obtain for q = 12 k − ( k − n inf(2 n,k ) X i =0 Γ ... × ki − i = R ( k )1 ,n = 2 n + 2 k − k X i =0 Γ ... × ki = 2 ( k +1) n (2.5)From the fact that the number of rank one persymmetric matrices over F is equal to three we obtain using combinatorial methods : NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK7 (2.6) Γ ... × k = (2 n − · The case n=5.
Set ( t , t , t , t , t ) = (cid:0) X i ≥ α (1) i T − i , X i ≥ α (2) i T − i , X i ≥ α (3) i T − i , X i ≥ α (4) i T − i , X i ≥ α (5) i T − i (cid:1) ∈ P . Denote by D × k ( t , t , t , t , t )the following 10 × k quintuple persymmetric matrix over the finite field F α (1)1 α (1)2 α (1)3 α (1)4 α (1)5 . . . α (1) k α (1)2 α (1)3 α (1)4 α (1)5 α (1)6 . . . α (1) k +1 α (2)1 α (2)2 α (2)3 α (2)4 α (2)5 . . . α (2) k α (2)2 α (2)3 α (2)4 α (2)5 α (2)6 . . . α (2) k +1 α (3)1 α (3)2 α (3)3 α (3)4 α (3)5 . . . α (3) k α (3)2 α (3)3 α (3)4 α (3)5 α (3)6 . . . α (3) k +1 α (4)1 α (4)2 α (4)3 α (4)4 α (4)5 . . . α (4) k α (4)2 α (4)3 α (4)4 α (4)5 α (4)6 . . . α (4) k +1 α (5)1 α (5)2 α (5)3 α (5)4 α (5)5 . . . α (4) k α (5)2 α (5)3 α (5)4 α (5)5 α (5)6 . . . α (4) k +1 We denote by Γ × ki the number of rank i quintuple persymmetric ma-trices over F of the above form :Let f ( t , t , t , t , t ) be the exponential sum in P defined by( t , t , t , t , t ) ∈ P −→ X degY ≤ k − X degU ≤ E ( t Y U ) X degU ≤ E ( t Y U ) X degU ≤ E ( t Y U ) X degU ≤ E ( t Y U ) X degU ≤ E ( t Y U ) . Then f k ( t , t , t , t , t ) = 2 k − rank (cid:2) D × k ( t ,t ,t ,t ,t ) (cid:3) JORGEN CHERLY
Hence the number denoted by R ( k ) q, of solutions( Y , U (1)1 , U (1)2 , U (1)3 , U (1)4 , U (1)5 , Y , U (2)1 , U (2)2 , U (2)3 , U (2)4 , U (2)5 , . . . Y q , U ( q )1 , U ( q )2 , U ( q )3 , U ( q )4 , U ( q )5 ) ∈ ( F [ T ]) q of the polynomial equations Y U (1)1 + Y U (2)1 + . . . + Y q U ( q )1 = 0 Y U (1)2 + Y U (2)2 + . . . + Y q U ( q )2 = 0 Y U (1)3 + Y U (2)3 + . . . + Y q U ( q )3 = 0 Y U (1)4 + Y U (2)4 + . . . + Y q U ( q )4 = 0 Y U (1)5 + Y U (2)5 + . . . + Y q U ( q )5 = 0 ⇔ U (1)1 U (2)1 . . . U ( q )1 U (1)2 U (2)2 . . . U ( q )2 U (1)3 U (2)3 . . . U ( q )3 U (1)4 U (2)4 . . . U ( q )4 U (1)5 U (2)5 . . . U ( q )5 Y Y ... Y q = satisfying the degree conditions degY i ≤ k − , degU ( i ) j ≤ , f or ≤ j ≤ ≤ i ≤ q is equal to the following integral over the unit interval in K Z P f qk ( t , t , t , t , t ) dt dt dt dt dt Observing that f ( t , t , t , t , t ) is constant on cosets of Q j =1 P k +1 in P the above integral is equal to(2.7) 2 q (10+ k ) − k +1) inf (10 ,k ) X i =0 Γ × ki − iq = R ( k ) q, where k > NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK9
We shall need the following results.
Result 1 :We have whenever k > " × ki = i = 0 ,
45 if i = 1 , · k + 1410 if i = 2 , · k + 31920 if i = 3 , · k + 42420 · k + 276640 if i = 4 , · k + 630000 · k − i = 5 , · k + 123480 · k − · k + 66170880 if i = 6 . · k − · k + 13332480 · k − i = 7 . · k − · k + 286720 · k − · k + 2 if i = 8 . where Γ " × ki denotes the number of quadruple persymmetric matrices in F of rank i (0 i inf(8 , k )) of the below form. α (1)1 α (1)2 α (1)3 α (1)4 α (1)5 . . . α (1) k α (1)2 α (1)3 α (1)4 α (1)5 α (1)6 . . . α (1) k +1 α (2)1 α (2)2 α (2)3 α (2)4 α (2)5 . . . α (2) k α (2)2 α (2)3 α (2)4 α (2)5 α (2)6 . . . α (2) k +1 α (3)1 α (3)2 α (3)3 α (3)4 α (3)5 . . . α (3) k α (3)2 α (3)3 α (3)4 α (3)5 α (3)6 . . . α (3) k +1 α (4)1 α (4)2 α (4)3 α (4)4 α (4)5 . . . α (4) k α (4)2 α (4)3 α (4)4 α (4)5 α (4)6 . . . α (4) k +1 Result 2
The Γ ... × ki where 0 ≤ i ≤ inf(2 n, k ) (see subsection 2.2 ) are solutionsto the below system. See Cherly[12 ] (2.9) Γ ... × k = 1 if k > ... × k = (2 n − · k > ... × k = 7 · n + (2 k +1 − · n − k +1 + 18 for k > ... × k = 15 · n + (7 · k − · n + (294 − · k ) · n −
176 + 14 · k for k > ... × k = 31 · n + 35 · k − · n + 2 k +2 − · k + 190286 · n +( − k +1 + 269 · k − · n + 2 k +2 − · k +2 + 94403 for k > ... × k = 63 · n + ( 1554 · k − · n + ( 52 · k − · k + 29150) · n + 12 · ( − · k + 6265 · k − · n + (35 · k − · k + 203872) · n − · k + 2960 · k − k > inf(2 n,k ) X i =0 Γ ... × ki = 2 ( k +1) n inf(2 n,k ) X i =0 Γ ... × ki − i = 2 n + k ( n − + 2 ( k − n − ( k − n − k inf(2 n,k ) X i =0 Γ ... × ki − i = 2 n + k ( n − + 2 − n + k ( n − · [3 · k −
3] + 2 − n + k ( n − · [6 · k − − − n + kn − · n ( k − − k + 8 · − n + k ( n − Result 3
The number of rank 10 quintuple persymmetric matrices of the form (1.1)is equal to :2 Y j =1 (2 k − − j ) . See Cherly[10, section 2 ]That is :
NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK11 (2.10) Γ × k = 2 Y j =1 (2 k − − j )2.4. Computation of the number of quintuple persymmetric ma-trices of the form (1.1) of rank I.Theorem 2.1.
We have whenever k > Γ × k = 1 if k > × k = 93 if k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = 1240 · [2 k + 3199 · k + 2 · · k − · ] for k > × k = 115320 · [2 k + 1148 · k − · · k + 311 · ] for k > × k = 496 · [2 k + 9525 · k − · k + 68115 · · k − · ] for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Proof.
Step 1
From the expressions of Γ " × ki in (2.8) we postulate that(2.12)Γ × ki = i = 0 ,a if i = 1 ,a · k + b if i = 2 ,a · k + b if i = 3 ,a · k + b · k + c if i = 4 ,a · k + b · k + c if i = 5 ,a · k + b · k + c · k + d if i = 6 .a · k + b · k + c · k + d if i = 7 .a · k + b · k + c · k + d · k + e if i = 8 .a · k + b · k + c · k + d · k + e if i = 9 .a · k + b · k + c · k + d · k + e · k + f if i = 10 . Step 2
Equally we postulate that :(2.13) Γ × k = 0 for k = 5Γ × k = 0 for k ∈ { , } Γ × k = 0 for k ∈ { , , } Γ × k = 0 for k ∈ { , , , } Γ × k = 0 for k ∈ { , , , , } NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK13
Step 3
Combining (2.9) with n=5 , (2.10) and (2.12) we obtain :(2.14) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = a · k + b · k + c · k + d for k > × k = a · k + b · k + c · k + d for k > × k = a · k + b · k + c · k + d · k + e for k > × k = a · k + b · k + c · k + d · k + e for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > and the relations:(2.15) X i =0 Γ × ki = 2 k +510 X i =0 Γ × ki − i = 2 k +5 + 1023 · k +510 X i =0 Γ × ki − i = 2 k +5 + 3162 · k +5 + 1045320 · k +5 Step 4
Computation of a , a in (2.14).From (2.14) and(2.15) we get: (cid:18) a a (cid:19) = · · + 1023 · · + 3162 · (2.16) ⇔ (cid:18) a a (cid:19) = (cid:18) (cid:19) Step 5
Computation of Γ × k in (2.14).From (2.13) and (2.16) we obtain :(2.17)Γ × k = a (2 k − )(2 k − )(2 k − )(2 k − ) = 31248 · [2 k − · k +71680 · k − · k +2 ] NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK15
To sum up we deduce from (2.17),(2.16) and (2.14)(2.18) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = a · k + b · k + c · k + d for k > × k = a · k + b · k + c · k + d for k > × k = 496 · k + b · k + c · k + d · k + e for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Step 6
Computation of a , a and b in (2.18).From (2.18) and (2.15) we get: a a b = · − · · · − · · · − · · (2.19) ⇔ a a b = · To sum up we deduce from (2.19),(2.18)(2.20) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = 1240 · k + b · k + c · k + d for k > × k = 115320 · k + b · k + c · k + d for k > × k = 496 · k + 496 · · k + c · k + d · k + e for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Step 7
Computation of c , d and e in (2.20).From (2.13) we get : NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK17 c d e = − · − · − · (2.21) ⇔ c d e = − · · − · Thus :Γ × k = 496 · k + 496 · · k − · · k + 2 · · k − · · [2 k + 9525 · k − · k + 68115 · · k − · ]To sum up we deduce from (2.22),(2.20) : (2.23) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = 1240 · k + b · k + c · k + d for k > × k = 115320 · k + b · k + c · k + d for k > × k = 496 · [2 k + 9525 · k − · k + 68115 · · k − · ] for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Step 8
Computation of b , b in (2.23).From (2.15) and (2.23) we get : (cid:18) b b (cid:19) = (2.24) ⇔ (cid:18) b b (cid:19) = (cid:18) · · (cid:19) NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK19
To sum up we deduce from (2.23),(2.24) :(2.25) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = 1240 · k + 1240 · · k + c · k + d for k > × k = 115320 · k + 115320 · · k + c · k + d for k > × k = 496 · [2 k + 9525 · k − · k + 68115 · · k − · ] for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Step 9
Computation of c , d in (2.25).From (2.13) we obtain : (cid:18)
32 164 1 (cid:19) (cid:18) c d (cid:19) = (cid:18) − · − · (cid:19) (2.26) ⇔ (cid:18) c d (cid:19) = (cid:18) − · · ( − · · · (2 · (cid:19) To sum up we deduce from (2.25),(2.26) :(2.27) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = 1240 · k + 1240 · · k + c · k + d for k > × k = 115320 · [2 k + 1148 · k − · · k + 2 · k > × k = 496 · [2 k + 9525 · k − · k + 68115 · · k − · ] for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Step 10
Computation of c in (2.27).From (2.15) and (2.27) we get :2 k · (cid:0)
62 + 6510 + 448260 + 22654800 + c + ( − · · · · + ( − · · (cid:1) = 0 ⇔ c = 2 · · · NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK21
To sum up we deduce from (2.28),(2.27) :(2.29) Γ × k = 1 for k > × k = 93 for k > × k = 31 · k +1 + 6386 for k > × k = 6510 · k + 364560 for k > × k = 620 · k + 448260 · k + 15748000 for k > × k = 65100 · k + 22654800 · k + 250817280 for k > × k = 1240 · k + 1240 · · k + 1240 · · · k + d for k > × k = 115320 · [2 k + 1148 · k − · · k + 2 · k > × k = 496 · [2 k + 9525 · k − · k + 68115 · · k − · ] for k > × k = 31248 · [2 k − · k + 71680 · k − · k + 2 ] for k > × k = 2 · [2 k − · k + 317440 · k − · k + 2080374784 · k − ] for k > Step 11
Computation of d in (2.29) :From (2.13) we deduce : Γ × = 0(2.30) ⇔ · + 1240 · · + 1240 · · · + d = 0 ⇔ d = − · [2 + 3199 · + 3913 · ] = − · · and Theorem 2.1 is proved. (cid:3) NUMERATION OF SOME PARTICULAR QUINTUPLE PERSYMMETRIC MATRICES OVER F BY RANK23
References [1] Landsberg, G Ueber eine Anzahlbestimmung und eine damit zusammenhangendeReihe, J. reine angew.Math, (1893),87-88.[2] Fisher,S.D and Alexander M.N. Matrices over a finite fieldAmer.Math.Monthly 73(1966), 639-641[3] Daykin David E, Distribution of Bordered Persymmetric Matrices in a finite field J.reine angew. Math, (1960) ,47-54[4] Cherly, Jorgen.Exponential sums and rank of persymmetric matrices over F arXiv : 0711.1306, 46 pp[5] Cherly, Jorgen.Exponential sums and rank of double persymmetric matrices over F arXiv : 0711.1937, 160 pp[6] Cherly, Jorgen.Exponential sums and rank of triple persymmetric matrices over F arXiv : 0803.1398, 233 pp[7] Cherly, Jorgen.Results about persymmetric matrices over F and related exponentials sumsarXiv : 0803.2412v2, 32 pp[8] Cherly, Jorgen.Polynomial equations and rank of matrices over F related to persymmetric matricesarXiv : 0909.0438v1, 33 pp[9] Cherly, Jorgen.On a conjecture regarding enumeration of n-times persymmetric matrices over F by rankarXiv : 0909.4030, 21 pp[10] Cherly, Jorgen.On a conjecture concerning the fraction of invertible m-times Persymmetric Matricesover F arXiv : 1008.4048v1, 11 pp[11] Cherly, Jorgen.Enumeration of some particular n-times Persymmetric Matrices over F by rankarXiv : 1101.2097v1, 18 pp[12] Cherly, Jorgen.Enumeration of some particular quadruple Persymmetric Matrices over F by rankarXiv : 1106.2691v1, 21 pp D´epartement de Math´ematiques, Universit´e de Brest, 29238 Brest cedex 3,France
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