${5choose 2}$ Proofs that ${nchoose k} leq {nchoose {k+1}}$ if $k<n/2$

 5 2  Pro ofs that  n k  ≤  n k +1  if k < n/ 2 . Dor on ZEILBERGER 1 There is no trivial mathematics, there are only tr ivial mathemat icians! A mathematician is trivial if he or she b eliev es that there exists tr ivial mathematics. But th is is not the only w a y for a mathematician to b e trivial. Another sufficien t condition for a m athematicia n to b e tr ivial is not to sho w up to a col lo quium talk w ith su c h an in triguing title and abstract! C on v ers ely , if yo u do sho w up, y ou are d efinitely non-trivial, so, congratulations, dear audience, y ou are the (only!) 19 non-trivial mathematicians in Columbia Univ ersit y . There are (at lea st) two ways to define  n k  . One wa y is  n k  := n ! k !( n − k )! . By this d efinition, the statemen t of the title is ind eed trivial:  n k   n k +1  = n ! k !( n − k )! n ! ( k + 1)!( n − k − 1)! = ( k + 1)!( n − k − 1)! k !( n − k )! = k + 1 n − k ≤ 1 , if k < n/ 2. The other definition is a c ombinatoria l one.  n k  is the n um b er of w a ys of cho osing a s et of k mem b ers out o f an n -element set. It is also the n um b er of n -letter words in the alphab et { S tr eet, Av enue } with exact ly k o ccurren ces of the “letter” “Street”, as I was reminded w hen I w alk ed, earlier to day , in th e (rea l!, not pro verbial) Manh attan latt ice from P ennsylv ania Station to the Columbia campus (except I confess that I c heated, and w alk ed m ost of the w a y on Broadwa y). If y ou adap t the latter definition, then  n k  = n ! k !( n − k )! b ecomes a the or e m , th at ca n b e pro v ed , e.g., by pr o ving that b oth sides satisfy th e recurr en ce (and initial cond ition). f ( n, k ) = f ( n − 1 , k − 1) + f ( n − 1 , k ) , f (0 , k ) = δ 0 ,k . (If after you w alk ed n blo cks, y ou are currently at the corner of k -th Street and ( n − k )-th Ave n ue, then one blo ck earlier, you were either at the corner of ( k − 1)-t h S treet and ( n − k )-th Ave n ue or the corner of k -th Street and ( n − k − 1)-th Av en u e). 1 Departmen t of Mathematics, Rutgers Univ ersi ty (N ew Brunswic k), Hill Center-Busc h Campus, 110 F relinghuyse n Rd., P iscata wa y , NJ 0 8854-8 019, USA. z eilber g at math dot rutgers do t e du , http:/ /www.m ath.rutg ers.edu/~zeilberg/ . March 4, 2010. A n almost verbatim transcript of a mathematics Colloq uium ta lk deli ve red at Colum bia Universi t y (Mathematics 5 2 0), F eb. 1 7, 20 10, 5 :00-6:00pm EST. I w ould lik e to thank Mikhail Khov a nov for inviting me, and the nineteen brav e souls who a ttended (f or exa mple W alter Neumann). I also w ould like to express my great disapp ointmen t at the 71 (faculty)+58 (grad studen ts)-19=110 p eop le (for example Da vid Ba y er) who did not attend. S u p po rted in part b y the NSF. Exclusiv ely published in the P ersonal Jo u rna l of Ekhad and Zeilberger ( http:// www.ma th.rutgers.edu/~zeilberg/pj.h tml ), and arx iv.org . 1 In particular, it follo ws that n ! k !( n − k )! is alwa ys an integ er!, whic h is not so ob vious, (since this is a r atio of t w o in tegers, that morally should b e a f raction, unless some miracle o ccurs), and that it is less than 2 n . T he sp ecial case that (2 n )! n ! 2 = (2 n )(2 n − 1) · · · ( n + 1) (1) · · · ( n ) is an in teger, and that it is less than 2 2 n , has an enormous num b er-theoretical significance. It w as u sed by Chebyc hev, in 1851, to “almost” prov e the Prime Numb er Theorem. Ev en though this b reakthrough was “sup erseded” b y the fu ll Prime Num b er T heorem, first p ro v ed at the end of the 19th-cen tury , all the pro ofs of the latter, as we ll as the later elementa ry p ro ofs of Erd˝ os and Selb erg, use C heb yc hev’s result as a stepping-stone for th e stronger statemen t. More r ecen tly , it turned out to b e crucial in the ama zing Agra w al-Ka y al-Saxena[AKS] P RI M E S ∈ P p ro of. Let’s recall Cheb ychev’s argumen t. S in ce p eople to d ay are so sp ecialized, I am willing to b et that man y of you ha ve nev er seen it b efore. Only this gem is worth the adm ission fe e of this talk (whic h is an hour of y our pr ecious time, th at at least % 80 of the Columbia facult y and graduate stud en ts found to o exorbitant.) Let’s lo ok at all th e prime n umbers b et ween n and 2 n . T hey must all d ivide the inte ger (2 n )! n ! 2 , so Y n ≤ p ≤ 2 n p ≤ 2 2 n , No w tak e log of b oth sides, define θ ( x ) = P p ≤ n log p , an d y ou w ould get that θ (2 n ) − θ ( n ) ≤ (2 n ) log 2 , that implies θ ( n ) − θ ( n/ 2) ≤ ( n ) log 2 θ ( n/ 2) − θ ( n/ 4) ≤ ( n/ 2) log 2 . . . Adding these up, y ou get that θ ( n ) ≤ (2 ln 2) n , w hic h is equiv alen t to π ( x ) ≤ C x log x for C = 2 ln 2 = 1 . 386 . . . . Later Chebyc hev made C ev en smaller, and Sylvester got v ery close to 1, and analogously for lo wer b ounds, b ut th e fu ll Pr im e Numb er Theorem had to wait for Hadamard and de la V all´ ee P oussin , in 1896. Going bac k to pro vin g (and reproving) that  n k  ≤  n k +1  if k < n/ 2, here is an i nductive pro of.  n k  =  n − 1 k − 1  +  n − 1 k  ≤  n − 1 k  +  n − 1 k + 1  =  n k + 1  , b y the induction hyp othesis , (pr o vid ed that the hyp othesis is fulfilled!). Th is is alwa ys true for  n − 1 k − 1  ≤  n − 1 k  , sin ce k < n / 2 implies k − 1 < ( n − 1) / 2, but for  n − 1 k  ≤  n − 1 k +1  it ma y happ en that 2 k < n / 2 but k ≥ ( n − 1) / 2. This happ en s exa ctly when n = 2 k + 1, and for this sp ecial case we ha v e to separately pr o ve:  2 k + 1 k  ≤  2 k + 1 k + 1  , but this f ollo ws f rom the ev en str onger fact that:  2 k + 1 k  =  2 k + 1 k + 1  , b y the sym m etry of the b inomial co efficien ts. I admit that th is is an ugly duckling of a pro of ( manipula torics , induction), but b y carefully traci ng it, we can get a b e autiful swan of a pro of, by defining an explicit injection that maps, in a c anonic al way , an n -letter word in the alphab et { S, A } with k S’s to one with k + 1 S’s. Simply look at the last time the num b er of Av en ues exceeded the num b er of Streets by exactly one, and sw ap Av en u es and Streets, un til then, and lea ve the r est in tact. The p ro of that I ju st ga v e is an example of a com binatorial pro of , a nd the pro cess of finding a c ombinatorial interpr etation to an algebraic identi t y or inequalit y , usin g a bijection and inj ection resp ectiv ely , is called C om bina torization . O ften algebraic/inductiv e pro ofs can b e “traced” and con v erted to b eautiful bijecti v e or in jectiv e pro ofs, lik e in the ab o v e case. Sp eaking of c ombinatorizat ion , this is the grand-daddy of a more recent trend, ca lled c ate gorific a- tion , made p opular b y m aster-blogge r John Baez. Categorificatio n b ecame a househo ld name when m y host, Mikhail Kho v ano v[Kh ], in 2000, astounded the mathematical world by c ate gorifying the famous Jones p olynomials, b y replacing a borin g p olynomial by an exciting c el l-c omplex . I strongly recommend Dror Bar-Natan’s ([B]) very lucid exp osition of Khov ano v’s seminal ideas, that yo u can easily fin d in . Going bac k to com binatorics, we will m eet other, ev en b etter, com b inatorial pro ofs, later on, bu t let me n o w presen t to y ou ye t another algebr aic pr o of . Using the Zeilb er ger algorithm [Z2] (or otherwise 2 ), we can find the gener ating function P n ( x ) := n X k =0  n k  x k = (1 + x ) n , and another w a y of stating that  n k  ≤  n k +1  if k < n/ 2 is to sa y that the coefficient s of P n ( x ) = (1 + x ) n first go up and then go down. Such a p olynomial is called unimo dal . In our case it is also symmetric . Let’s call a symmetric and unimo dal p olynomial with non-negativ e intege r co efficien t a Z -p olynomial, and let’s call the dar ga o f P its lo w -degree plu s its (high)-degree. F or example, the dar ga of x 4 + x 5 is 9 wh ile th e dar ga of x 3 is 6. 2 Algebra is really com binatorics in disguise, when y o u expa nd (1 + x ) n you make n independent decisions, whether to pick the 1 or the x . The co efficien t of x k is the the n um b er of ways of c ho osing which k of the n terms will donate it s x to the common cause. 3 The follo wing t w o simple facts (tak en from m y de-c ombinatorization ([Z1]) of Kath y O’Hara’s ([O]) seminal com binatorial p ro of of the unimo dalit y of th e Gaussian p olynomials) are easily pro ved. F act 1 : The sum of t w o Z-p olynomials of the same d arga is another Z-p olynomial of that darga. F act 2 : The p ro duct of t w o Z-p olynomials is y et-another-one, and its darga is the s um of their dargas. T o pro ve F act 2 note th at the add itiv e “ato ms” o f Z -p olynomials are p olynomials of the form x i + x i +1 + ... + x j and multiplying out t w o suc h atoms would yield ( x a + x a +1 + ... + x b )( x c + x c +1 + ... + x d ) = x a + c +2 x a + c +1 +3 x a + c +2 + . . . + 3 x b + d − 2 +2 x b + d − 1 + x b + d , whic h is in deed a Z-p olynomial of darga ( a + c ) + ( b + d ) = ( a + b ) + ( c + d ). It follo ws immediately , by induction, that (1 + x ) n is a Z-p olynomial, since 1 + x is. But we ge t, for the same p rice, that man y other p olynomials are Z -p olynomials, and hence automaticall y unimo dal. F or examp le ( x + x 2 + x 3 + x 4 + x 5 + x 6 ) n , whic h has the follo wing probabilistic in terpretation. Y ou roll a fair die n times and at eac h roll you win as man y dollars as the n um b er of dots that sho w up. Then y ou are more lik ely to win k + 1 dollars than k dollars as long as k is less than your exp ected gain 7 n/ 2 . More generally: (1 + x ) m ( x + x 2 + x 3 + x 4 + x 5 + x 6 ) n ( x + x 2 + x 3 + x 4 ) k , that also has a gam b ling inte rpretation, and many more complicated gambling scenarios, th at y ou are wel come to mak e u p. Let’s tak e a closer lo ok at the ab o v e com binatorial pro of that  n k  ≤  n k +1  , that consisted in defining an explicit injection b et w een k -sets to ( k + 1)-sets. It inputs a set S with k elemen ts ( k < n / 2) and o utputs a set with one more elemen t by lo oking at the sm allest inte ger r suc h that | S ∩ { 1 , 2 , . . . , r }| = ( r − 1) / 2 and map p ing it to the s et ( { 1 , . . . , r }\ S ) ∪ ( S ∩ { r + 1 , . . . , n } ). F or examp le, with n = 11 and k = 4 the 4-set { 1 , 2 , 4 , 11 } is mapp ed to the 5-set { 3 , 5 , 6 , 7 , 11 } (in this example r = 7). Not e th at for this injection the outpu t-set do es not con tain the input set. It would b e more d esirable, an d natur al , if w e could come-up with an injection S → S ′ from the collect ion of k -sets to the collectio n of k + 1-sets that has the prop ert y that S ⊂ S ′ , in other w ords, find a “rule” that adds a new mem b er to S , as long as k < n/ 2, and in suc h a wa y that no t w o differen t S s w ould give the same S ′ . 4 If th er e w ould b e suc h a mapping w e w ould get, b y iterating it, a maximal chain that ends at the middle r ank. B y symm etry , if w e r eflect this to the complement, we wo uld get a c e ntr al chain de c omp osition of the Bo olean lattice (a lias n -dimensional unit cub e). Conv ersely , an y suc h c hain- decomp osition of th e Bo olean lattice w ould give suc h an injection, and would yield y et-another-pro of of the un imo dalit y of the binomial coefficients. The easiest wa y to construct su c h a c h ain decomp osition is r e cursively . T ake any sym metric c h ain of B n − 1 C r → C r +1 → . . . → C n − 1 − r , and construct t wo new c hains in B n . The fi rst is the same C r → C r +1 → . . . → C n − 1 − r , but viewe d as b elonging to B n , and the second is C r ∪ n → C r +1 ∪ n → . . . → C n − 1 − r ∪ n . There is only one p roblem! Neither c hains are le gitimate symmetric chains in B n . The sum o f the starting rank and ending rank (in the case of the Boolean lattice, the rank of a set is its num b er of elemen ts) s h ould b e n , whereas the first c h ain has the sum to o low , namely n − 1, w h ile the second c h ain h as its rank to o high , namely n + 1. T o get t w o n ew c hains that are just right , we cut th e last mem b er of th e second chain and put it at the end of the first c hain, getting the t w o c hains: C r → C r +1 → . . . → C n − 1 − r → C n − 1 − r ∪ n , and C r ∪ n → C r +1 ∪ n → . . . → C n − 2 − r ∪ n . Let’s illustrate this constru ction for n ≤ 3. F or n = 1 we only ha ve one c hain, namely: ∅ → { 1 } . This giv es rise to tw o c hains for n = 2: ∅ → { 1 } → { 1 , 2 } , { 2 } . The firs t of these gives rise to tw o c hains for n = 3: ∅ → { 1 } → { 1 , 2 } → { 1 , 2 , 3 } , { 3 } → { 1 , 3 } , while the sin gleton c hain { 2 } on ly gi v es r ise to one c h ain (the seco nd one is empt y) { 2 } → { 2 , 3 } . 5 Martin Aigner came up with another w a y of constru cting a symmetric c hain decomp osition for the Boolean latti ce B n , that may b e termed lexic o gr aphic gr e e d . S tart with the empt y set, and at eac h lev el lo ok at the lexicographically fir s t set that has not y et b een committed and that con tains the current tail of the emerging c h ain. Keep doing it unt il y ou get stuc k. Su rprisingly , you get a symmetric c hain d ecomp osition. Wh y?, b ecause it happ ens to b e the same as the one ab ov e. So ev en though man y p eople would find Aigner’s construction more elegan t and app ealing, the easiest w a y to pr ov e its v alidit y is to discov er the recursive construction ab o v e and then it is easy to p ro v e b y induction that it is indeed th e same. The d ra wbac k that b oth the recurs ive and Aigner’s([A]) lexicographic-greed appr oac hes sh are is that y ou ha v e to construct al l chains , and find out how the injection acts on al l sets, at on ce, requiring exp onent ial time and space. What if you only care ab out the successor of just one individual set? C u rtis Greene and Daniel Kleitman[GK] came up with a v ery elega n t d escription of (essent ially the same!) injection. There is a one-to-one m apping b et ween sets and words in the alphab et { [ , ] } . F or any set S of n elemen ts form the “wo rd” ( w 1 , . . . , w n ) by the r ule w i = [ iff i ∈ S . F or example, the empt y set for n = 4 corresp onds to the word ]]]] and th e whole set { 1 , 2 , 3 , 4 } corresp ond s to the word [[[[. If y ou ha v e a le gal br acketing then it forms its o wn singleton chain. Otherwise, “compile” it to the b est of y our abilit y , matc hing a left-brack et “[” w ith a righ t one “]”. Once y ou ha v e finished “compiling” yo u w ould get a b unch of ]’s follo wed by a b unch of [’s which is as illegal as it gets, p ossibly (and us ually) in tersp ersed with clusters of legal brac k etings. Lea ve these legal brac k etings alone, and c hange the last ] by a [. In sym b ols: L 1 ] L 2 ] L 3 ] . . . L k − 1 ] L k ] L k +1 [ L k +2 [ . . . [ L r go es to L 1 ] L 2 ] L 3 ] . . . L k − 1 ] L k [ L k +1 [ L k +2 [ . . . [ L r If y ou can’t do it (i. e. k = 0), then the c hain ends. The existence of a symmetric c hain decomp osition for the Bo olean lattice immediately implies Sp erner’s theorem that the largest p ossible collection of s u bsets of { 1 , 2 , . . . n } such that none of its mem b ers pr op erly cont ains another one (what is called an anti-chain , or c lutter ) equals  n [ n/ 2]  . Ob viously , this is sharp, since the collec tion of al l [ n/ 2]-sets , that has  n [ n/ 2]  mem b ers, is ob viously an ant i-c h ain. Can y ou do b etter? Of course not! Th e n um b er of symmetric chains in an y s ymmetric c hain decomp osition of the Bo olean lattice (and we kn ow that one exists) equ als  n [ n/ 2]  , sin ce eac h c h ain p asses once through the middle-rank, and ev ery set b elongs to exactly one c h ain. Giv en any an ti-c hain, there can b e at most one-set-p er-c hain, or else it w ould not b e an an ti-c hain! While the ab o ve pr o of of Sp erner’s theorem is my p ersonal fa v orite, let me remind y ou of another, just-as-nice pr o of fr om the b o ok , due to Da vid Lu b ell[L]. 6 There are n ! p ossib le chains that start at the top, the empt y set, and end -up at the b ottom ( { 1 , 2 , . . . n } ) (in obvio us one-one corresp ondence with p erm utations). Let C b e a p otent ial an ti- c h ain. F or eac h S ∈ C , there are exactly | S | !( n − | S | )! suc h top-to-bottom c hains that p ass through S , and of course, no tw o different mem b ers of C can share such a top-to-b ottom c hain, or else they w ould b e r elated! So w e hav e the obvious inequalit y X S ∈C | S | !( n − | S | )! ≤ n ! , that implies that X S ∈C 1  n | S |  ≤ 1 . But the maxim um of  n | S |  is  n [ n/ 2]  (thanks to the main th eorem of the pr esen t article!), so the minim um of 1 /  n | S |  is 1 /  n [ n/ 2]  , and w e hav e |C |  n [ n/ 2]  ≤ 1 , as claimed. The L ast (and longest! (yet the b est!)) Pro of W e ha ve already presen ted ab o v e sev eral c ombinatorial pro ofs of  n k  ≤  n k + 1  , if k < n/ 2 , b y fi nding a set-the or etic al injection b et ween the collection of k -sets and th e collect ion of ( k + 1)- sets, i.e. b et w een t w o sets (of set s). In general, a c ombinato rial pro of of a ≤ b consists of constructing sets A and B such a = | A | and b = | B | , and an injection f : A → B . But, there is yet another way , a linear-algebra pro of ! Come-up with t w o ve ctor sp ac es A and B suc h that dim ( A ) = a a nd dim ( B ) = b and co nstruct a line ar tr ansformation T : A → B , and prov e th at T is an injection b y pro ving that for any f ∈ A , T f = 0 implies f = 0. 7 Let V k b e the v ector space spanned b y all k -subsets of { 1 , . . . , n } , in other words the v ector space of all “formal sums” (as they would say in algebraic top ology) X | S | = k a S S , where a S are mem b ers of your fa vo rite field (sa y the fi eld of rational num b ers, or ev en GF ( p ) for an y prime p large r th an n ). Our p rop osed m apping, M : V k → V k +1 , so on to b e prov ed an injection, is defined on basis elemen ts b y M ( S ) = X j 6∈ S ( S ∪ j ) , and extended linearly . What is the “meaning” of M ( S )? Supp ose that yo u enlarge your current facult y S b y another memb er, and y ou can’t decide, and y ou wa n t to hire ev ery one who is not already in S but yo u are only allo wed to hire one p erson. If yo u liv e in a classical world, you w ould ha v e to make- up your mind , make one new professor happy , but disapp oint all the other applican ts. But in the quan tum world, you can ha v e a “sup erp osition” of all scenarios for “hiring an extra pr ofessor”. In ord er to pr ov e that M is indeed an in jection, w e n eed a “companion op erator”: L : V k → V k − 1 , defined on basis elemen ts b y: L ( S ) = X i ∈ S ( S \ i ) , and extended linearly . L ( S ) has an analogous meaning in a quantum w orld. Because of bu dget cuts, you ha ve to fire one pr ofessor, bu t you don’t wan t to get any one upset, so y ou ha v e a q u an tum- sup er p osition of a ll firing-one-professor s cenarios. I no w claim that on V k , ML − LM = µ ( k ) I , where µ ( k ) = 2 k − n ( n is fixed through this pro of ), and I is the iden tity mapp ing. O f course, b y linearit y , it is enough to pro ve th is for b asis elemen ts S ∈ V k : ML ( S ) − LM ( S ) = µ ( k ) S . (1) ML ( S ) is formal sum of all scenarios of fi re-and-then-hire wh ile LM ( S ) is the form al sum of all scenarios of hire-and-then -fi re. If the guy you h ir ed an d the guy y ou fired are differ e nt then “hire- Smith-then-fire-Jones” yiel ds the same set as “fire-Jones-then-hire-Smith” and s o they ca ncel out. The only scenarios that do not cancel out are those where the guy you fired and the guy you hired are one an d the same. There are k w a ys to fi re-and-then-hire th e same p erson, and there are n − k w a ys to hire-and-then-fi re the same p erson, at eac h case r esulting in th e original set S . Th is giv es a net con tribution of k − ( n − k ) = 2 k − n copies of S . 8 Next I claim that on V k , for any r ≥ 1 ML r − L r M = ( µ ( k ) + . . . + µ ( k − r + 1)) L r − 1 . (2) This follo ws easily b y induction on r , by u s ing ML r +1 − L r +1 M = ( ML r − L r M ) L + L r ( ML − L M ) . So, if f ∈ V k , w e ha v e: ( ML r +1 − L r +1 M ) f = ( ML r − L r M )( L f ) + L r ( ML − L M ) f . (3) Since L f ∈ V k − 1 , we ha ve from the ind uction h yp othesis that th e first term on the righ t side of (3) equals ( ML r − L r M )( L f ) = ( µ ( k − 1) + . . . + µ ( k − r )) L r − 1 ( L f ) = ( µ ( k − 1) + . . . + µ ( k − r )) L r f , (3 a ) and since ( ML − LM ) f = µ ( k ) f , the second term of (3) is L r ( ML − LM ) f = L r µ ( k ) f = µ ( k ) L r f . (3 b ) Incorp orating (3 a ) and (3 b ) in to (3), we get: ( ML r +1 − L r +1 M ) f = ( ML r − L r M )( L f ) + L r ( ML − L M ) f = ( µ ( k − 1) + . . . + µ ( k − r )) L r f + µ ( k ) L r f = ( µ ( k ) + . . . + µ ( k − r )) L r f , that is (2) with r replaced b y r + 1. No w supp ose th at there is an f ∈ V k suc h that M f = 0. W e ha v e to prov e that f = 0. By (2) we ha v e that ML r f = ( µ ( k ) + . . . + µ ( k − r + 1)) L r − 1 f . Applying M r − 1 to b oth sides giv es M r L r f = ( µ ( k ) + . . . + µ ( k − r + 1)) M r − 1 L r − 1 f . Iterating, giv es: M r L r f = ( µ ( k ) + . . . + µ ( k − r + 1))( µ ( k ) + . . . + µ ( k − r + 2)) · · · ( µ ( k )) f . So w e hav e M k +1 L k +1 f = ( N on − Z er o − N umber ) f . But, since f ∈ V k , L k f is a multiple of the empty set, and hence L k +1 f = 0 ( L∅ = 0 , since in that case w e get the empty sum in the d efinition of L ∅ ). So w e get that f = 0, as promised. 9 So indeed, if k < n/ 2, th e mapping M : V k → V k +1 is an injection, and w e get dim ( V k ) ≤ dim ( V k +1 ), and s o, once again, we kno w th at  n k  ≤  n k +1  if n < k / 2. But wh y w ork so hard, if we had th e f ormer far easier pro ofs? On e reason, is why not ? Who said that an elegan t p ro of h as to b e short? Another reason is that this pro of extends , almost v erbatim, to other lattices, for whic h no simp le pro ofs of ran k -u nimo dalit y and the Sp ern er pr op ert y are kno wn. The pro of that I just present ed w as in spired by , and is along similar lines as-but n ot quite the same- as Rob ert Pro ctor’s ([P]) b eautiful sim p lification of Richard Stanley’s([S]) seminal p ro of of the Sp erner prop ert y for lattices of in teger partitions. T h e main part in the pro of of Sp ernerit y , pro ving th at M is injectiv e (as w e ju s t did), can b e traced, in an almost equiv alen t f orm (bu t using differen tial op erators op erating on so-called semi-in v arian ts) to James Joseph Sylvest er[Sy] w a y bac k in 1878. References [A] Martin Aigner, L exic o gr aphic matching in Bo ole an algebr as , J. Com binatorial Theory Ser. B 14 (1973), 187–194. [AKS] Manindra Agra wal, Neera j Ka yal and Nitin Saxena, PRIM ES is in P , Annals of Mathematics, Second Series 160 (200 4), 78 1-793 . [B] Dror Bar-Nat an, On Khovanov’s c ate gorific ation of the Jones p olynomial , Algebr. Geom. T op ol. (electronic) 2 (2 002), 337–370 . [GK] Curtis Greene and Daniel J . Kleitman, Pr o of te chniques in the the ory of finite sets , Studies in combinato rics, MAA Stu d. Math. 17 , (G. -C. Rota, ed.), Math. Asso c. America, W ashington, D.C., 1978 [K] Mikhail Kho v ano v , A c ate gorific ation of the J ones p olynomial , Duke Math. J. 101 (2000), 359–4 26 . [L] Da vid Lu b ell, A short pr o of of Sp erner’s lemma , J. Com binatorial Theory 1 (1 966), 299. [O] Kathleen M. O’Hara, U nimo dality of Gaussian c o efficients: a c onstructive pr o of , J. Combin. Theory Ser. A 53 (1990), 29–52. [P] Rob ert A. Pro ctor, Solution of two difficult c ombinatorial pr oblems with line ar algebr a , Amer. Math. Mon thly 89 (1982), 72 1–734 [St] Ric hard P . Stanley , Weyl gr oups, the har d L efschetz the or em, and the Sp erner pr op erty , SIAM J. Algebraic Discrete Methods 1 ( 1980), 16 8–184 . [Sy] James J oseph Sylveste r, Pr o of of the hitherto undemonstr ate d fundamental the or em of invari- ants , Philosophical Magazine 5 (187 8), 178-188 . (Also in Collected W orks v. 3, 117 -126, Chelsea, NY, 1973). 10 [Z1] Doron Zeilb erger, A one-line high scho ol algebr a pr o of of the u nimo dality of the Gaussian p olynomial s [ n k ] for k < 20. In : q -series and partitions (Minneap olis, MN, 1988), D. Stant on, ed., 67–72 , IMA V ol. Math. App l., 18 (1989), S pringer, New Y ork. [Z2] Doron Zeilb erger, The metho d of cr e ative telesc oping , J. Symb olic Comput. 11 (1991), 195–204 . 11