THE SPLITTING SUBSPACE CONJECTURE

Transcription

1 THE SPLITTING SUBSPAE ONJETURE ERI HEN AND DENNIS TSENG Abtract We anwer a uetion by Niederreiter concerning the enumeration of a cla of ubpace of finite dimenional vector pace over finite field by proving a conjecture by Ghorpade and Ram Introduction We poitively reolve the Splitting Subpace onjecture, temming from a uetion poed by Niederreiter (995) [3, p and tated by Ghorpade and Ram [2 We firt define the notion of a σ-plitting ubpace Definition In the vector pace F mn over the finite field F, given a σ F mn uch that F mn F (σ), a (m-dimenional) ubpace W of F mn i a σ-plitting ubpace if W σw σ n W F mn For example, {, σ m, σ 2m,, σ (n )m} pan a σ-plitting ubpace If n, then F mn i the only σ-plitting ubpace; if m, then each -dimenional ubpace of F mn i σ-plitting onjecture (Ghorpade-Ram) The number of σ-plitting ubpace i mn m m(m )(n ) Thi follow a orollary 34 from our main reult, Theorem 33 The next two ection are devoted to proving thi theorem We firt contruct a recurion that give the cardinality of more general clae of ubpace, including the σ-plitting ubpace, and then olve thi recurrence to obtain the reult Finally, we dicu ome pecial cae of our more general reult 2 Recurion For the remainder of thi report, unle otherwie noted, conider more generally the vector pace F N ( F N ) over the finite field F, given a σ F N uch that F N F (σ) We begin by iolating the key property of the linear tranformation v σv Propoition 2 The linear endomorphim of F N that preerve no ubpace other than {0} and all of F N are exactly thoe which act a multiplication by a primitive element σ that generate the extenion F (σ) F N Proof Operator defined a multiplication by a primitive element σ generating the extenion F (σ) F N cannot preerve any ubpace except {0} and F N, for if W i uch a ubpace with nonzero w W, then w N i0 a iσ i W, a i F, o W F N onverely, note that any linear operator T together with the vector pace F N can be viewed a a finitely generated

2 2 ERI HEN AND DENNIS TSENG F [x module M, where x act a T Since F [x i a principal ideal domain, we can ue the primary decompoition of M to find M k i F [x/(p i (x) r i ), where p i i a polynomial for each i and r i i a poitive integer If T preerve no proper ubpace of F N, then k Alo, r unle p (T )M i a proper ubmodule of M Therefore, we have M i eual to F [x/(p (x)), where p i an irreducible polynomial Thi i exactly what it mean for x( T ) to act a the primitive element of the field extenion F (σ) F N F N with minimal polynomial p (x) We next define notation to decribe the et to be counted by the general recurion Definition Suppoe that A, A 2,, A k are et of ubpace of F N Let [A, A 2,, A k be the et of all k-tuple (W, W 2,, W k ) uch that W i A i for i k, W i W i+ + σw i+ for i k If A i i the et of all ubpace of F N with dimenion d i, then A i i denoted within the bracket a d i For example, [3, A 2 denote all tuple (W, W 2 ) uch that dim(w ) 3, W 2 A 2 and W W 2 + σw 2 Definition For nonnegative integer a, b with N > a > b or a b 0 (a, b) : { W F N : dim(w ) a and dim(w σ W ) b } For example, (, 0) i the et of all -dimenional ubpace and (2, ) i the et of all 2-dimenional ubpace W uch that dim(w σ W ) Definition Given et [A,, A,2, [A 2,, A 2,2,, [A r,, A r,2 a defined above, let [A,, A,2, [A 2,, A 2,2,, [A r,, A r,2 denote the et of 2r-tuple of ubpace (W,, W,2, W 2,, W 2,2, W r,, W r,2 ) uch that (W i,, W i,2 ) [A i,, A i,2 for i r, W i,2 W i+, for i r For example, [3, 2, [2, i the et of all 4-tuple of ubpace (W, W 2, W 3, W 4 ) uch that dim(w ) 3, dim(w 2 ) 2, dim(w 3 ) 2, dim(w 4 ), W W 2 + σw 2, W 3 W 4 + σw 4, W 2 W 3 We ue the following propoition extenively in contructing the recurion Propoition 22 For nonnegative integer N > a > b or a b 0 [a, b max(a,0) ib max(b,0) j0 [(a, i), b [a, (b, j) Proof Follow from Propoition 2 and the definition of [,, (, )

3 THE SPLITTING SUBSPAE ONJETURE 3 We next define an ordering on the tuple labelling the et of ubpace [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) The recurion in Lemma 23 will give the cardinality of et of ubpace o labelled in term of the cardinality of et labelled by tuple before it in the ordering The bae cae i [(0, 0), containing one element Definition Firt, define an ordering on the ordered pair of the form (a, b) uch that (a, b ) (a 2, b 2 ) if a > a 2 or a a 2 and b < b 2 Next, define an ordering on tuple of the form [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) uch that the order i lexicographic in term of the ordered pair (a i,, a i,2 ) from left to right Finally, define an ordering on the ame tuple for 0 uch that [(a,, a,2 ), (a 2,, a 2,2 ),, (a r+,, a r+,2 ) [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) For example, (3, ) (3, 2) (2, 0) and [(6, 5), (4, 2) [(6, 5), (4, 3) [(5, 2), (2, 0) Lemma 23 Suppoe N > a, > a,2 a 2, > a 2,2 a r, > a r,2 0 a r+, a r+,2 a r+, a r+,2 and after etting a 0, a 0,2 N, a r+, a r+,2 0, j r+ k r+ 0, that (or ele [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) i empty) Let Then a i, 2a i, a i,2 for i r {(j,, j r ) : max(a i+,2, 2a i,2 a i, ) j i max(a i,2, 0), i r}, D {(k,, k r ) : a i,2 k i a i,, i r} [(a,, a,2 ), (a 2,, a 2,2 ),, (a r+,, a r+,2 ) [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) [ ai,2 (2a i,2 j i ) [(a,2, j ), (a 2,2, j 2 ),, (a r,2, j r ) a (j,,j r) i i, (2a i,2 j i ) [ ki a i+, [(a,, k ), (a 2,, k 2 ),, (a r,, k r ) a i,2 a i+, (k,,k r) D\(a,2,,a r,2 ) Proof We give an example before the general cae Let r 2; we compute [(3, ), (, 0) by counting [3,, [, 0 in two different way Applying Propoition 22 to the term on the left within the bracket give [3,, [, 0 [(3, 2),, [(, 0), 0 + [(3, ),, [(, 0), 0 Above, if (W, W 2, W 3, W 4 ) [(3, 2),, [(, 0), 0 then W 2 W 3 and W 4 {0} So [(3, 2),, [(, 0), 0 [(3, 2), (, 0) Likewie for (W, W 2, W 3, W 4 ) [(3, ),, [(, 0), 0 then W 2 W 3 and W 4 {0} So [(3, ),, [(, 0), 0 [(3, ), (, 0), i

4 4 ERI HEN AND DENNIS TSENG and [3,, [, 0 [(3, 2), (, 0) + [(3, ), (, 0) Next, applying Propoition 22 to the term on the right within the bracket give [3,, [, 0 [3, (, 0), [, (0, 0) If (W, W 2, W 3, W 4 ) [3, (, 0), [, (0, 0), then W 3 W 2, W 4 {0} and thu W i a 3-dimenional ubpace containing the 2-dimenional pace W 2 + σw 2 So [ N 2 [3, (, 0), [, (0, 0) [(, 0), (0, 0) and therefore [ N 2 [3, (, 0), [, (0, 0) [(, 0), (0, 0) We then have, after rearranging, that [ N 2 [(3, ), (, 0) [(, 0), (0, 0) Note that [(3, 2), (, 0), [(, 0), (0, 0), [(3, 2), (, 0) come before [(3, ), (, 0) in the ordering on tuple The proof of the Lemma i a generalization of thi proce The firt euality i clear The ize of [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) i computed by applying Propoition 22 (R) [a,, a,2, [a 2,, a 2,2,, [a r,, a r,2 [(a,, k ), a,2, [(a 2,, k 2 ), a 2,2,, [(a r,, k r ), a r,2 (k,,k r) D (k,,k r) D Expanding in the other way, we get (L) [(a,, k ), (a 2,, k 2 ),, (a r,, k r ) [ ki a i+, i a i,2 a i+, [a,, a,2, [a 2,, a 2,2,, [a r,, a r,2 [a,, (a,2, j ), [a 2,, (a 2,2, j 2 ),, [a r,, (a r,2, j r ) (j,,j r) (j,,j r) [(a,2, j ), (a 2,2, j 2 ),, (a r,2, j r ) Subtracting from (R) and (L) the uantity produce the tated reult of the Lemma i [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) [ ai,2 (2a i,2 j i ) a i, (2a i,2 j i ) Finally, we relate et of the form [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) to σ-plitting ubpace

5 Propoition 24 Let F N THE SPLITTING SUBSPAE ONJETURE 5 F mn Then [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) { } n 2 n 3 n ( σ i W, σ i W,, W σw, W ) : σ i W F mn i0 i0 In particular [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) i the number of σ-plitting ubpace Proof If W i a σ-plitting ubpace, then n 3 σ i W, σ i W,, W σw, W ) n 2 ( i0 i0 [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) On the other hand, uppoe that (W n,, W ) [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) Then for k n 2 dim(w k+ ) (k + )m 2km (k )m dim(w k + σw k ) So W k+ W k + σw k for k n 2 Alo, W 2 W σw a W σw {0} Suppoe that W k k i0 σi W Then, ince dim(w k+ ) dim(w k + σw k ) (k + )m, we obtain W k+ W k + σw k i0 W + σ 2 W + + σ k W k σ i W i0 When k n, we have that W n + σw n n i0 σi W, ince W n + σw n F mn i mn-dimenional So W i indeed a σ-plitting ubpace orollary 25 The number of σ-plitting ubpace in F mn over F i independent of choice of primitive element σ Proof Neither the bae cae [(0, 0 nor Lemma 23 depend on the σ choen Remark More generally, given an arbitrary invertible linear operator T on F mn over F, we might conider how many T -plitting ubpace exit; that i, the number of m-dimenional ubpace W uch that W T W T n W F mn We may then redefine (, ), [,,, by replacing the expreion W + σw with W + T W and W σ W with W T W Recall from Propoition 2 and Lemma 23 that when T (v) σv, the nonzero number [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) can be computed from the bae cae [0, 0

6 6 ERI HEN AND DENNIS TSENG But if T i any invertible linear operator, there may exit nonempty et of the form [(a, a ), (a 2, a 2 ),, (a r, a r ) where a r 0 In fact, uch et cannot be computed recurively For example [4, 4, [2, 2 [(4, 4), 4, [(2, 2), 2 [(4, 4), (2, 2) [4, (4, 4), [2, (2, 2) [(4, 4), (2, 2) We may till apply Lemma 23 in the cae of general T, however, with the cardinalitie of thee et a additional bae cae Remark One might try to apply the method in Lemma 23 to try to count pointed ubpace Namely, given a fixed vector v, it i not difficult to how that the number of tuple in [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ) whoe l th ubpace contain v i eual to a l, N [(a,, a,2 ), (a 2,, a 2,2 ),, (a r,, a r,2 ), and the number of tuple (W,, W,2,, W r,, W r,2 ) in [a,, a,2, [a 2,, a 2,2,, [a r,, a r,2 uch that W l, contain v i eual to a l, N [a,, a,2, [a 2,, a 2,2,, [a r,, a r,2 ounting thee tuple (W,, W,2,, W r,, W r,2 ) in [a,, a,2, [a 2,, a 2,2,, [a r,, a r,2 by expanding at a,, a 2,,, a r, a in (R), we get exactly a l, N (k,,k r) D [(a,, k ), (a 2,, k 2 ),, (a r,, k r ) [ ki a i+, i a i,2 a i+, which i jut (R) caled by al, It i le obviou that expanding at a N,2,, a r,2 will reult in (L) caled by al, onidering two cae, one where the ubpace W N l, +σw l, contain v and the other where the ubpace W l, + σw l, doe not contain v, yield the expanion (j,,j r) [ 2al,2 j l [ al,2 (2a l,2 j l ) a l, (2a l,2 j l ) + ([ a l,2 [(a,2, j ), (a 2,2, j 2 ),, (a r,2, j r ) [ 2a l,2 j l i,i l, )[ a l,2 (2a l,2 j l ) a l, (2a l,2 j l ) [ ai,2 (2a i,2 j i ) a i, (2a i,2 j i ) It can be checked that [ 2al,2 j l [ al,2 (2a l,2 j l ) a l, (2a l,2 j l ) + ([ a l,2 [ 2a l,2 j l )[ a l,2 (2a l,2 j l ) a l, (2a l,2 j l ) [ al, [ al,2 (2a l,2 j l ) a l, (2a l,2 j l ), which mean that we again jut get the expanion (L) caled by a l, N

7 THE SPLITTING SUBSPAE ONJETURE 7 3 Solution to the Recurion The next two lemma are pecial cae of the following -hu-vandermonde identity for N a nonnegative integer [, p 354 Refer to Appendix A for proof of the lemma ( ) 2φ N, a; c N /a N ( ) ( N ; ) m (a; ) m c N m : c (; ) m (c; ) m a m0 (c/a; ) N (c; ) N Lemma 3 If B D A are non-negative integer, then B [ [ [ [ A B B A (2B ) B D (2B ) [B [ [ [ B A B D [D D B B (B )(B ) Lemma 32 If D B A are non-negative integer, then B [ [ [ [ A B B (B )(B ) B D D [B [ [ [ B B A D [A D D B D We now give the main theorem of thi report Theorem 33 Suppoe that () Then where N > a, > a,2 a 2, > a 2,2 a r, > a r,2 0, [(a,, a,2 ),, (a r,, a r,2 ) E a 0, a 0,2 N, a r+, a r+,2 0 [ a, r [ ai, a i+, i0 a i+, a i+,2 r i r (a i, a i,2 )(a i, a i,2 ) i [ ai+, [ ai, a i+, a i+,2 [ ai+,2 a i+2, orollary 34 (Splitting Subpace onjecture) We have, when N mn, the euality [ N mn + m [((n )m, (n 2)m),, (2m, m), (m, 0) [ m m In particular, when N mn, [((n )m, (n 2)m),, (2m, m), (m, 0) [ mn [ m m(m )(n ) E, m(m )(n )

8 8 ERI HEN AND DENNIS TSENG Proof From plugging into () [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) ( [N (n )m [m ) ( [(n )m [m ) [ (n )m [ (n )m m (n )m Since for each k n 2, [ (n )m [ m m [ m m m [ (n )m (n 2)m [ (n )m (n 2)m [ (n )m (n 2)m [ (n k)m (n k )m [ (n k)m (n k )m [ N (n )m m [ N mn + m m [m 0 (n 2)m [2m m 0 [m m(m )(n ) [(n k)m! [m![(n k )m! [(n k)m! [m![(n k )m! 0 n i m(m ) (n k)m (n k )m, thi reduce to (n )m m m(m )(n ) m(m )(n ) Proof of Theorem 33 We verify that () atifie the recurion in Lemma 23 Recall that L [ ai,2 (2a i,2 j i ) [(a,2, j ), (a 2,2, j 2 ),, (a r,2, j r ) a (j,,j r) i i, (2a i,2 j i ) [ ki a i+, R [(a,, k ), (a 2,, k 2 ),, (a r,, k r ) a i,2 a i+, (k,,k r) D We firt check euality when a r,2 0 o that the expreion obtained for (L) and (R) uing () do not contain negative -binomial Subtituting () and applying Lemma 3 to the reulting independent um in (L) give L R [ a,2 [ a,2 [ a, r i j i [ ai,2 a i,2 a i,2 j i i [a i,2 [a i, a i+,2 i [ ai,2 [ ji [ ai,2 (2a i,2 j i ) j i a i+,2 a i, (2a i,2 j i ) (a i,2 j i )(a i,2 j i ) r [ ai,2 i a i+,2 [ ai,2 a i+,2 [ ai,2 a i,2 a i, a i,2 [ ai, a i+,2 a i,2 a i+,2 r i, [ ai,2 a i+,2 Subtituting () and applying Lemma 32 to the reulting independent um in (R) give r [ ai, a i, [ ai, [ ki [ ki a i+, i k i a i, k i a i,2 a i+, [ a, i [a i, [a i, a i,2 a i+, k i r i a i+, [ ai,2 a i+,2 a i,2 a i+, (a i, k i )(a i, k i ) [ [ [ ai, ai, a i+, ai, a i,2 a i, a i,2 r i [ ai, a i+,

9 (2) After implification (ee Appendix B) L R ( N )[N a,2! [N a,! [a r,2! THE SPLITTING SUBSPAE ONJETURE 9 i [a i, a i+,2! [a i, a i,2! [a i, a i,2! i2 [a i,2 a i,! Finally, we deal with the cae a r,2 0, when the expreion obtained by directly applying () to (L) may contain negative -binomial ((R) i unaffected) Suppoe r > By definition, we know that [a,, a,2, [a 2,, a 2,2,, [a r,, 0 [a,, a,2, [a 2,, a 2,2,, [a r,, a r,2 Thi mean that L (j,,j r) [(a,2, j ), (a 2,2, j 2 ),, (a r,2, j r ) [a,, a,2,,, [a r,, a r,2 [ ar,2 Since a r,2 a r, > 0, we may apply our previou reult to obtain [ ar,2 [a,, a,2,, [a r,, a r,2 [ ar,2 a r, a r, ( N )[N a,2! [N a,! [a r,2! We wih to how that thi i eual to ( N )[N a,2! [N a,! [a r,2! i r when a r,2 0 Take the uotient to find i a r, i [a i, a i+,2! [a i, a i,2! [a i, a i,2! [a i, a i+,2! [a i, a i,2! [a i, a i,2! ( N )[N a,2! r [a i, a i+,2! r [N a,! [a r,2! i [a i, a i,2! [a i, a i,2! i2 [ ar,2 ( a r, N )[N a,2! r [a i, a i+,2! r [N a,! [a r,2! i [a i, a i,2! [a i, a i,2! i2 [a r,2! [a r,2! [a r,2!, [a r, a r+,2! [a r, a r,2! [a r, a r,2! [a r,2 a r,! [ar,2 a r, [a r,2! [a r,! [a r,! [a r,! [a r,2 a r,! [ar,2 a r, [a r,! [a r,2 a r,! [ar,2 a r, [ ai,2 (2a i,2 j i ) a i, (2a i,2 j i ) i2 r i2 [ ar,2 a r, [a i,2 a i,! [a i,2 a i,! [a i,2 a i,! [a i,2 a i,!

10 0 ERI HEN AND DENNIS TSENG a deired Therefore, when a r,2 0, the euality L (j,,j r) [[a,2, j ), [a 2,2, j 2 ),, [a r,2, j r ) ( N )[N a,2! [N a,! [a r,2! R i till hold Finally, uppoe r Then, [ N L [(0, 0) [ ai,2 (2a i,2 j i ) a i i, (2a i,2 j i ) [a i, a i+,2! [a i, a i,2! [a i, a i,2! [a i,2 a i,! a, [ N If we plug in [a,, 0 into (2), then we get a,, a deired orollary 35 The number are given by L R ( N )[N a,2! [N a,! [a r,2! a, [a,, a,2, [a 2, a 22,, [a r,, a r,2 i [a i, a i+,2! [a i, a i,2! [a i, a i,2! 4 Special ae: (k,k-) i2 i2 [a i,2 a i,! Note that when r and a, k, a,2 k, with k N, the formula () give [ N (k, k ) a number independent of k Propoition 4 There i a bijection between et of the form (k, k ) and (k 2, k 2 ) when k, k 2 N Proof It uffice to how that there exit a bijection between (k, k ) and (k, k 2) for 2 k N Define The map φ i well defined:, φ : (k, k 2) (k, k ) W W + σw dim(w + σw ) k, dim((w + σw ) (σ W + W )) k The econd euality follow from the fact that (W + σw ) (σ W + W ) contain W and ha dimenion trictly le than k by Propoition 2 Next, φ i injective: if W, W 2 (k, k 2) and W +σw W 2 +σw 2 W (k, k ), then W σ W W W 2 Finally, φ i urjective: if W (k, k ), then W σ W (k, k 2) ince (W σ W ) + σ(w σ W ) W ; in fact (W σ W ) + σ(w σ W ) W

11 THE SPLITTING SUBSPAE ONJETURE 5 A analogue In light of our reult, we might ak what () count when we et In thi ection we will ee that the ituation tranlate from enumerating ubpace of vector pace to enumerating ubet of et Intead of ubpace of F N, we conider ubet of {,, N} Rather than multiplying by the element σ, we let σ cyclically permute the element of {,, N}, o that σ preerve no proper ubet, in analogy with Propoition 2 onverely, it i eay to ee that any permutation of the et {,, N} that preerve no proper ubet i cyclic, and we can reorder the element o that σ (2 N) in cycle notation For example, we have σ{, 3, 4} {2, 4, 5} for N 5 When N mn, the number of m-element ubet W of {,, N} uch that n i0 σi W {,, N} i eaily een to be n Propoition 5 will how thi in a lightly more general etting We retain the [,, (, ), <, > notation a before with the definition retated in the etting of ubet of {,, N} below Definition Suppoe A, A 2,, A k are et of ubet of {,, N} Let [A, A 2,, A k be the et of all k-tuple (W, W 2,, W k ) uch that W i A i for i k, W i W i+ σw i+ for i k If A i i the et of all ubet of {,, N} with cardinality d i, then A i i denoted within the bracket a d i Definition For nonnegative integer a, b with N > a > b or a b 0 (a, b) : { W F N : W a, W σ W b } Definition Given et [A,, A,2, [A 2,, A 2,2,, [A r,, A r,2 a defined above, let [A,, A,2, [A 2,, A 2,2,, [A r,, A r,2 denote the et of 2r-tuple of ubet (W,, W,2, W 2,, W 2,2, W r,, W r,2 ) uch that (W i,, W i,2 ) [A i,, A i,2 for i r, W i,2 W i+, for i r It can be checked that Lemma 23 i till valid in thi etting, o our formula are till valid by jut plugging in Thi occur ince the -binomial counting way to extend ubpace become binomial term counting way to enlarge ubet However, we can directly count ome pecial cae and check that they agree with the general formula Propoition 5 (Analogue of the Splitting Subpace onjecture) For and N mn we have [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) N ( ) N mn + m m m In particular, if N mn, then [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) n

12 2 ERI HEN AND DENNIS TSENG Proof The ame argument ued in Propoition 24 can be ued to how that if (W n,, W ) [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0), then W i+ W i σw i Therefore, the problem of counting thi et reduce to finding the number of m-element ubet W uch that W σw σ n W ha cardinality mn We count the number of ordered pair (W, k), where W i an m-element ubet atifying the deired property and k i an element of W Firt, we fix k and count the number of et W that contain k Without lo of generality, uppoe k N n + Since W cannot have any element between N 2n + 2 and N n incluive, chooing the ret of W amount to chooing m element from {,, N 2n + } uch that the element are at leat n apart By an elementary counting argument, there i a bijection between chooing m element from {,, N 2n + } uch that the element are at leat n apart and chooing m element from an N 2n + (m 2)(n ) N mn + m element et Thi i exactly ( ) ( N mn+m m Therefore, the number of ordered pair (W, k) i N N mn+m ) m, ince there are N choice of k We can alo count the number of ordered pair (W, k) by chooing W firt and then k Since there are m poibilitie once we fix W, each W appear in m ordered pair So a deired [((n )m, (n 2)m), ((n 2)m, (n 3)m),, (2m, m), (m, 0) N ( ) N mn + m m m In particular, if N mn then mn m ( mn mn+m ) m n Remark A, the formula in Propoition 5 agree with the formula in orollary 34 ( Propoition 52 If, then (m, k) N N m ) ( N m m k N N m m ) m m k )( k )( m k Proof We count in two way the ordered pair (W, a) where W i an m-element et uch that W σw k and a / W Firt, fix a and count the number of poible W not containing a Without lo of generality, uppoe a N Then, define a block of W {,, N } to be a ubet {b, b +,, b + l } of conecutive number contained in W uch that b, b + l / W (if b, then b i undertood to be N, which i already fixed to not be in W ) We know that the um of the ize of the block of W i m, ince the union of the block i m Alo, if B,, B i are the block of W, then we know that ( B ) + + ( B i ) k, ince the interection of W with σw i preciely the dijoint union of all the block of W without the firt element of each block In particular, the number of block mut be m k Therefore, to count the number of poibilitie of W, we count the number of way to pace out the block of W : the number of way to chooe m k element out of (N ) k conecutive number uch that no two element are adjacent Thi i euivalent to the number of way to chooe m k element out of an N k (m k ) N m element et Now that we have fixed the pacing of the block, the number of way to ditributing the remaining k element of W into the m k block i ( ) (m k )+k k Therefore, the number of

13 THE SPLITTING SUBSPAE ONJETURE 3 )( m poible ordered pair (W, a) i N ( ) N m m k k, ince there are N choice for the initial value of a We can alo count the number of ordered pair (W, a) by fixing W and then finding the number of poibilitie for a For a fixed W there are N m poible a Thi mean that (m, k) N N m ( N m m k )( m k ) N m k ( N m m k )( m k ) N m Remark For a power of a prime, () how that [ [ N m m (m, k) [ m m k k Thi give the ame anwer a Propoition 52 when ( N m m k )( m k ) (m k)(m k ) Appendix A Proof of Lemma 3 and 32 Proof of Lemma 3 Firt ubtitute B for to find that the identity i euivalent to B [ [ [ [ A B B B A B (+) 0 B D B [B [ [ [ B A B D [D D B B Then [ A B ( A B ) ( A B ) ( ) ( ) ( ) (A B ) ( 2) (B+ A ; ) (; ), and [ [ B B B (; ) B (; ) B (; ) + (; ) B (; ) (; ) B (; ) B ( )( 2 ; ) (; ) (; ) B ( B ) ( B ) ( B ) ( )( 2 ; ) (; ) [ B ( B ) ( B ) ( B ) ( )( 2 ; )

14 4 ERI HEN AND DENNIS TSENG Finally [ B ( B ) ( )( 2 ; ) [ A B D B [ A B D B [ A B D B [ A B D B [ A B D B [ A B D B [ A B D B (;) A B (;) D B (;) A D (;) A B (;) D B (;) A D ( D B ) ( D B ) ( A B ) ( A B ) ( ) (D B ) ( 2) ( B+ D ; ) ( ) (A B ) ( 2) ( B+ A ; ) ombining thee term [ [ A B B [ B B [ [ B B A B D B B [ B [ A B D B The power of i (D ) ince ( ) (A B ) + (B ) 2 ( ) + (D B ) ((A B ) 2 (D ) Thi mean that B [ [ A B B [ B 0 B [ [ B B A B D B 0 [ [ [ B B A B D B ( ) (B ) ( 2) ( + B ; ) [ A B (+) D B (D ) (B+ A ; ) ( + B ; ) ( B+ D ; ) (; ) ( 2 ; ) ( B+ A ; ) (D ) (+ B ; ) ( B+ D ; ) (; ) ( 2 ; ) B ( ) 2 ( ) ) + ( + ) 2 [ A B (+) D B ( + B ; ) ( B+ D ; ) (D ) (; ) ( 2 ; ) ( ) 2φ + B, B+ D ; D 2

15 a deired THE SPLITTING SUBSPAE ONJETURE 5 [ [ [ B B A B ( D+ B ; ) B D B ( 2 ; ) B [ [ [ B B A B ( D+ B ) ( D ) D B ( 2 ) ( B ) [ [ [ [ B B A B D D B B D [ [ [ B B A B D D D B B Proof of Lemma 32 The proof i identical to that of Lemma 3 Subtitute B for to find that the lemma i euivalent to B D [ [ [ [ A B B B B (+) 0 B D [B [ [ [ B B A D [A D D B D We ee that [B D [ B D (;) B (;) D (;) B D (;) B (;) D (;) B D [ B D [ B D [ B D [ B D ( B D ) ( B D ) ( B ) ( B ) (B D ) ( 2) ( D+ B ; ) (B ) ( 2) ( + B ; ) ombining thi with the expreion for [ A B and [ B B Lemma 3 [ [ [ A B B B B D B B [ [ B B D [ [ B B D [ B [ B (+) from the proof of (A D) (B+ A ; ) ( + B ; ) ( D+ B ; ) (; ) ( 2 ; ) ( + B ; ) (A D) (B+ A ; ) ( D+ B ; ) (; ) ( 2 ; )

16 6 ERI HEN AND DENNIS TSENG The power of i (A D) becaue ( ) ( ) (A B ) + (B ) 2 2 ( ) ( ) + (D + B) ((B ) ) + ( + ) 2 2 (A D) Thi mean that B D [ [ A B B [ [ B B 0 B D [ [ B B D B B D 0 [ [ B B B D a deired B B B B A D (+) ( B+ A ; ) ( D+ B ; ) (A D) (; ) ( 2 ; ) ( ) 2φ B+ A, D+ B ; A D 2 [ [ B B ( B+ D ; ) A B D ( 2 ; ) A B [ [ B B ( B+ D ) ( A D ) D ( 2 ) ( A B ) [ [ [ B B A D D A B A D [ [ [ B B A D D B D Proof L r i [ a,2 i [N! [N! [! [a,2! [a,2! [! [a i,2! [a i,2! [! Appendix B Proof of Theorem 33, LR [a i,2 [a i, a i+,2 [a r,2! [a r,2! [! [a r,! [a r,! [! [ ai,2 a i+,2 [ ar,2 a r,2 a r, a r,2 [a i+,2! [a i,2 a i+,2! [a i,2! [a i, a i+,2! [a i, a i+,2! [! [ [ ai,2 a i,2 ai, a i+,2 a i, a i,2 a i,2 a i+,2 [a r,! [a r,2! [a r, a r,2! [ ai,2 a i,2 a i, a i,2 r i [ ai,2 a i+,2 [a i, a i+,2! [a i,2 a i+,2! [a i, a i,2! [a i,2! [a i+,2! [a i,2 a i+,2!

17 THE SPLITTING SUBSPAE ONJETURE 7 [N! [ [N! ar,2 a r,2 [a,2! [a r,! [a r,2! a r, a r,2 [a,2! [a r, a r,2! r i [a i,2! [a i,2! [a i, a i+,2! [a i+,2! [a i,2 a i+,2! [a i,2 a i,2! [a i, a i,2! [a i,2 a i,! [a i+,2! [a i, a i,2! [N! [ [N! ar,2 a r,2 [a,2! [a r,! [a r,2! a r, a r,2 [a,2! [a r, a r,2! r i [a i,2! [a i+,2! [a i+,2! [a i,2! [a i,2 a i,2! [a i,2 a i+,2! [a i, a i+,2! [a i, a i,2! [a i,2 a i,! [a i, a i,2! [N! [N! [a r,2 a r,2! [a,2! [a r,! [a r,2! [a r, a r,2! [a r,2 a r,! [a r, a r,2! [a,2! [a,2! [a r,2! [N a,2! [a r,2! [a,2! [a r,2 a r,2! ( N )[N a,2! [a r,2! ( N )[N a,2! [N a,! [a r,2! i r i [a i, a i+,2! [a i, a i,2! [a i,2 a i,! [a i, a i,2! [a i, a i+,2! [a i, a i,2! [a i,2 a i,! [a i, a i,2! [a i, a i+,2! [a i, a i,2! [a i, a i,2! [a i,2 a i,! i i2 R r i [ a, i [N! [N! [! [a,! [a,! [! [a i,! [a i,! [! [a i, [a i, a i,2 [a r,! [a r,! [! [ [ [ ai, ai, a i+, ai, a i,2 a i+, a i,2 a i+, [a r,! [a r, a r,2! [a r, a r,2! [! [a i,! [a i, a i,2! [a i, a i,2! [! a i, a i,2 r i [a r, a r,2! [a r,2! [a r, a r,2! [a r, a r,2! [a r, a r,! [a i, a i+,! [a i+,! [a i, a i+,! [a i,2 a i+,! [a i, a i,2! [N! [N! [a,! [a,! [a r,! [a r,2! [a r, a r,2! [a r, a r,2! [a r, a r,2! [a r, a r,! r i [a i,! [a i, a i,2! [a i, a i,2! [a i+,! [a i,2 a i+,! [a i, a i,2! [N! [N! [a,! [a,! [a r,! [a r,2! [a r, a r,2! [a r, a r,2! [a r, a r,2! [a r, a r,! r i [ ai, a i+, [ ai, a i,2 a i, a i,2 [ ai, a i+, [a i, a i,2! [a i, a i,2! [a i, a i,! [a i,! [a i+,! [a i, a i+,! [a i,! [a i+,! [a i+,! [a i,! [a i, a i+,! [a i, a i,! [a i, a i,2! [a i,2 a i+,! [a i, a i,2! [a i, a i,2!

18 8 ERI HEN AND DENNIS TSENG [N! [a,! [a r,! [a r, a r,2! [N! [a,! [a r,2! [a r, a r,2! [a r, a r,2! [a r, a r,! [a,! [a r,! [a r, a r,! r [a i, a i,2! [a r,! [a,! [N a,! [a i i,2 a i+,! [a i, a i,2! [a i, a i,2! [N! [a r, a r,2! [N! [a r,2! [a r, a r,2! [a r, a r,2! [a r,! r [N a,! i ( N ) [a r,2! [a r,! [N a,! ( N )[N a,2! [N a,! [a r,2! [a i, a i,2! [a i, a i,2! [a i, a i,2! [a i2 i,2 a i,! [a i, a i,2! [a i i, a i,2! [a i, a i,2! [a i2 i,2 a i,! [a i, a i+,2! [a i, a i,2! [a i, a i,2! i i2 [a i,2 a i,! Acknowledgement Thi reearch wa conducted at the 202 ummer REU (Reearch Experience for Undergraduate) program at the Univerity of Minneota, Twin itie, and wa upported by NSF grant DMS and DMS We would like to thank Prof Gregg Muiker, Pavlo Pylyavkyy, Vic Reiner,and Denni Stanton, who directed the program, for their upport, and expre particular gratitude to Prof Stanton both for introducing u to thi problem and for hi indipenable guidance throughout the reearch proce We alo thank Alex Miller for hi aitance in editing thi report Reference [ G Gaper and M Rahman, Baic Hypergeometric Serie, Enc of Math and it Appl, Vol 96, ambridge Univerity Pre, ambridge, 2004 [2 S R Ghorpade and S Ram, Enumeration of plitting ubpace over finite field, Arithmetic, Geometry, and oding Theory, (Luminy, France, March 20), Y Aubry, Ritzenthaler, and A Zykin Ed, ontemporary Mathematic, Vol 574, American Mathematical Society, Providence, RI, 202, 0pp, to appear [3 H Niederreiter, The multiple-recurive matrix method for peudorandom number generation, Finite Field Appl (995), 3-30 Eric hen, Princeton Univerity, Princeton, NJ addre: ecchen@princetonedu Denni Teng, Maachuett Intitute of Technology, ambridge, MA addre: DenniTeng@gmailcom