Zeros in the character table of the symmetric group

ZER OS IN THE CHARA CTER T ABLE OF THE SYMMETRIC GR OUP SARAH PELUSE AND KANNAN SOUND ARARAJAN T o T r evor Wo oley on the o c c asion of his sixty first birthday Abstract. Computations of Miller and Sc heinerman suggest that the v ast ma jorit y of the zeros appearing in the c haracter table of the symmetric group are of a certain sp ecial type. While we cannot prov e this, we resolve a conjecture arising in their pap er concerning these zeros, and address a related question of Stanley . 1. Introduction In [7] w e sho w ed that almost all entries in the c haracter table of the symmetric group S N are m ultiples of an y given prime n um b er p , and in [8] extended this to divisibility b y prime p o w ers. This resolv ed conjectures of Miller [5]. It is conceiv able that these divisibility results hold b ecause most of the c haracter v alues are in fact zero, as is the case for high rank finite simple groups of Lie type [4]. This p ossibility has not b een ruled out, but the n umerical evidence (first generated by Miller [5], who computed the c haracter table for N ≤ 38) suggests that the prop ortion of zeros in the character table tends to zero with N . More recen t large-scale Monte Carlo simulations by Miller and Scheinerman [6] suggest m uc h more refined conjectures concerning the zeros in the c haracter table. In this pap er w e address a conjecture arising in their work, and also a related question of Stanley [9]; the problem of determining an asymptotic for the n umber of zeros in the c haracter table remains op en, but we can determine the num b er of zeros pro duced by three w ell-known criteria for guaran teeing zeros. In order to state these conjectures fluidly , w e review some notation and facts concerning the c haracter table of the symmetric group. Let λ and µ denote partitions of the integer N . W e denote by χ λ µ the v alue of the irreducible c haracter of S N corresp onding to the partition λ ev aluated on the conjugacy class of p erm utations in S N with cycle type corresp onding to the partition µ . It is an immediate consequence of the Murnaghan–Nak ay ama rule (see [3, Theorem 2.4.7]) that if µ has a part of size t and λ is a t -core, then χ λ µ = 0. Miller and Sc heinerman call the pair of partitions ( λ, µ ) corresp onding to a zero χ λ µ = 0 in the character table a zer o of typ e II if λ is a t -core for some part size t of µ . A pair ( λ, µ ) corresp onds to a zer o of typ e I if λ is a µ 1 -core, where µ 1 denotes the largest part of µ . Clearly , type I zeros are a prop er subset of type I I zeros. The work of the first author was partially supp ored by NSF grant DMS-2516641 and a Sloan Research F ello wship and the work of the second author was partially supp orted by NSF grant DMS-2100933. 1 2 SARAH PELUSE AND KANNAN SOUNDARARAJAN The computations of Miller and Sc heinerman suggest that most of the zeros app earing in the c haracter table of S N are of t yp e I I, and that almost all t yp e I I zeros are t yp e I zeros. After seeing the first v ersion of their preprint, w e pointed out to them that it is easy to sho w that the prop ortion of t yp e I zeros in the character table of S N is asymptotically 2 log N . Based on this and their computational data, Miller and Sc heinerman made the follo wing conjecture. Conjecture 1 (Miller and Sc heinerman) . The pr op ortion of entries e qual to zer o in the char acter table of S N is asymptotic al ly 2 log N . There is a third sufficient condition for χ λ µ to equal zero, which includes all the zeros of t yp e I I. F or an y natural n um b er t , let H t ( λ ) denote the n um b er of hook lengths in the Y oung diagram of λ that are multiples of t . F urther, define P t ( µ ) = 1 t X t | µ j µ j , so that t P t ( µ ) is the sum of the parts in µ that are m ultiples of t . Then a criterion of Stanley [10] (see Lemma 7.4) states that (1) χ λ µ = 0 if P t ( µ ) > H t ( λ ) for some integer t. W e will call zeros ( λ, µ ) for which the criterion in (1) holds zer os of typ e III . W e b ecame aw are of this criterion through a comment of Stanley on MathOverflo w [9]. There, Stanley drew atten tion to the follo wing elegant corollary of the criterion (1), whic h ma y also b e found as Corollary 7.5 of [10] or as Exercise 7.60 of [11]: the c haracter v alue χ λ µ equals zero if, in the p olynomial ring Z [ x ], (2) ℓ ( µ ) Y i =1 (1 − x µ i ) ∤ Y u ∈ λ (1 − x h ( u ) ) . Here ℓ ( µ ) equals the n um b er of parts in µ , whic h w e denote in descending order by µ 1 , . . . , µ ℓ ( µ ) and h ( u ) denotes the ho ok length corresp onding to the box u in the Y oung diagram of λ . By considering the ro ots of b oth sides of (2), one may see the following equiv alen t form u- lation of the criterion (2): χ λ µ = 0 if there is an in teger t such that the n um b er of parts in µ that are m ultiples of t exceeds the num ber of ho oks in λ that are m ultiples of t . Since P t ( µ ) is at least the num ber of parts in µ that are multiples of t , we see that condition (2) is a corollary of (1). F urther, if λ is a t -core, then H t ( λ ) = 0, and so if µ has a part equal to t , then (2) implies that χ λ µ = 0. This rev eals that zeros of type I I I include the zeros of type I I previously describ ed (which in turn include the zeros of type I). Our main result determines an asymptotic for the n um b er of zeros of types I, I I, or I I I. It turns out that all three types of zeros hav e the same leading order asymptotic, so that most zeros of type I I I are in fact simply zeros of type I. ZER OS IN THE CHARA CTER T ABLE OF THE SYMMETRIC GROUP 3 Theorem 1. F or a natur al numb er N , let Z I ( N ) , Z I I ( N ) , and Z I I I ( N ) denote the numb er of p airs ( λ, µ ) c orr esp onding to zer os of typ e I, II, and III r esp e ctively. Then for inte gers N ≥ 3 , al l thr e e quantities Z I ( N ) , Z I I ( N ) and Z I I I ( N ) ar e asymptotic al ly p ( N ) 2  2 log N + O  (log log N ) 2 (log N ) 2  , wher e p ( N ) denotes the numb er of p artitions of N . Mor e over, Z I I I ( N ) − Z I I ( N ) = O ( N − 1 2 p ( N ) 2 ) . This result resolves a conjecture of Miller and Sc heinerman that most zeros of type I I are in fact of t yp e I. It also addresses a question of Stanley [9], who asked for an understanding of the n um b er of t yp e I I I zeros, and we see that these are very nearly equal to the n um b er of type I I zeros. It would b e p ossible to give a more accurate asymptotic for Z I ( N ), with sharp er error terms: namely , with an error term O ( N − c p ( N ) 2 ) for a suitable p ositiv e constant c , Z I ( N ) is approximately p ( N ) 2 Z 1 0 exp  − y  1 + log √ 6 N π y  dy . This can b e expressed as an asymptotic expansion p ( N ) 2 log( √ 6 N /π ) ∞ X k =0 P k (log log √ 6 N /π )(log √ 6 N /π ) − k , where P k is a p olynomial of degree k . F or small N , the effect of the lo w er order terms in our asymptotic for Z I ( N ) is noticeable: for instance, with N = 50000, the precise asymptotic expansion ab ov e suggests that the prop ortion of type I zeros is ab out 0 . 13 whic h is in go o d agreemen t with the empirical data in [6], whereas the leading order term in Theorem 1 gives a substantially larger prop ortion of ab out 0 . 1848. The bulk of the difference b etw een Z I I ( N ) and Z I ( N ) is comprised of pairs ( λ, µ ) where λ is a µ 2 -core but not a µ 1 -core. T o determine the asymptotic coun t of suc h pairs, we would need to understand how differen t ho ok lengths are distributed in a random partition λ , which w e do not know. 2. Preliminaries on the asymptotic for p ( N ) The n um b er of conjugacy classes in S N , whic h equals the num ber of irreducible representa- tions of S N , is giv en by the partition function p ( N ). In this section w e recall some standard facts regarding the Hardy–Ramanujan asymptotic form ula for p ( N ), namely p ( N ) ∼ exp(2 π √ N / √ 6) 4 N √ 3 . Both the asymptotic formula and the salient features of the pro of that we now recall will b e useful in our subsequent w ork. 4 SARAH PELUSE AND KANNAN SOUNDARARAJAN F or | z | < 1 denote the generating function for p ( N ) by F ( z ) = ∞ Y n =1 (1 − z n ) − 1 = ∞ X N =0 p ( N ) z N . The Hardy-Ramanujan form ula is obtained through an analysis of the Cauc hy integral for- m ula applied to F ( z ): (3) p ( N ) = 1 2 π i Z | z | = q F ( z ) z − N − 1 dz = 1 2 π Z π − π F ( q e iθ ) q − N e − iN θ dθ , where the parameter q < 1 is c hosen carefully . The v alue q is c hosen so as to minimize F ( r ) r − N o v er all r ∈ (0 , 1). Note that log F ( z ) = ∞ X n =1 log(1 − z n ) − 1 = ∞ X n =1 ∞ X k =1 z nk k = ∞ X m =1 σ ( m ) m z m , up on grouping terms according to m = nk and writing σ ( m ) = P m = nk n . Th us to minimize F ( r ) r − N , we must c ho ose r so that r F ′ ( r ) F ( r ) = N , or in other words, ∞ X m =1 σ ( m ) r m = N . A suitable v alue of q , which is very nearly the optimal solution to the abov e equation, is giv en b y (4) q = exp  − π √ 6 N  , and throughout this pap er when w e write q we ha ve this choice in mind. F or large x , a straigh tforw ard asymptotic calculation gives (5) ∞ X n =1 σ ( n ) n e − n/x = π 2 6 x − 1 2 log(2 π x ) + O ( x − 1 ) . In fact, one can b e m uc h more precise thanks to a mo dularity relation but (6) will be sufficien t for our purp oses. With q as in (4) it follows that (taking x = √ 6 N /π in (5)) (6) log F ( q ) = ∞ X m =1 σ ( m ) m exp  − π m √ 6 N  = π √ 6 √ N − 1 2 log(2 √ 6 N ) + O ( N − 1 2 ) . Returning to the righ t-most formula in (3), the integrand drops sharply in size as | θ | increases, and the contribution from | θ | substan tially larger than N − 3 4 is negligible. T o see this, note ZER OS IN THE CHARA CTER T ABLE OF THE SYMMETRIC GROUP 5 that log |F ( q e iθ ) | F ( q ) = − ∞ X m =1 σ ( m ) m (1 − cos mθ ) q m ≤ − ∞ X m =1 q m (1 − cos mθ ) = −  q 1 − q − Re q e − iθ 1 − q e iθ  , and a small calculation (in which it is helpful to distinguish the cases | θ | ≤ (1 − q ) and π ≥ | θ | > (1 − q )) establishes that for an absolute constant c > 0, (7) log |F ( q e iθ ) | F ( q ) ≤ − c min( N 3 2 θ 2 , N 1 2 ) . F rom our analysis ab o v e, w e conclude that (8) p ( N ) = 1 2 π Z | θ |≤ N − 3 5 F ( q e iθ ) q − N e − iN θ dθ + O  F ( q ) q − N exp( − cN 3 10 )  . The choice (4) of q is suc h that, for small θ , the argument of F ( q e iθ ) e − iN θ is approximately zero and F ( q e iθ ) e − iN θ is approximated b y F ( q ) e − C N 3 2 θ 2 for a suitable absolute constant C > 0. Ev aluating the resulting Gaussian in tegral leads to the Hardy–Ramanujan formula. W e record t wo further useful facts following from this analysis: (9) p ( N ) ∼ F ( q ) q − N 2 · 6 1 4 · N 3 4 ∼ exp(2 π √ N / √ 6) 4 N √ 3 and (10) Z π − π |F ( q e iθ ) | q − N dθ ≪ N − 3 4 F ( q ) q − N ≪ p ( N ) . 3. Distribution of hook lengths F or an y partition λ , the ho ok asso ciated to a b ox u in the Y oung diagram of λ is the collection of b oxes to the right of u in the same row as u , b elow u in the same column as u , and u itself. The length of this ho ok, denoted by h ( u ), equals the num b er of b oxes in the ho ok. Giv en a partition λ of N and a natural n umber t we are in terested in the quantit y H t ( λ ), whic h denotes the num ber of ho ok lengths in λ that are multiples of t . W e will require a result of Han [2] that giv es a useful formula for the generating function for H t ( λ ). If z and w are complex num bers inside the unit circle, Theorem 1.3 of [2] gives (11) ∞ X N =0 X λ ⊢ N z N w H t ( λ ) = ∞ Y n =1 (1 − z n ) − 1 (1 − z tn ) t (1 − w n z tn ) t . A partition λ of N with H t ( λ ) = 0 (thus no ho ok lengths that are m ultiples of t ) is known as a t -core partition, and the n umber of such partitions is denoted by c t ( N ). Han’s formula 6 SARAH PELUSE AND KANNAN SOUNDARARAJAN generalizes the well-kno wn formula for the generating function for t -cores, (12) ∞ X N =0 c t ( N ) z N = ∞ Y n =1 (1 − z n ) − 1 (1 − z tn ) t . In this section, we shall show that H t ( λ ) is suitably large for most partitions λ in v arious ranges of t . T o describe these results fluidly , it is con venien t to define the follo wing thresholds for t : (13) T 0 = √ 6 N π (log N ) − 4 , T 1 = √ 6 N π  log √ N − 20  , and (14) T 2 = √ 6 N π  log √ N + log log N − log log log N − 20  . Only the thresholds T 0 and T 1 are relev an t for this section, and the threshold T 2 will only app ear in Section 5 when we complete the proof of the main theorem. Throughout w e assume that N is sufficiently large, so that quan tities lik e log log log N are well defined. Prop osition 1. L et N b e lar ge, and r e c al l that q = exp( − π / √ 6 N ) . (i) If t ≤ T 0 is an inte ger, then ther e ar e ≪ N − 10 p ( N ) p artitions λ of N with H t ( λ ) ≤ N t  1 − 2 log N  . (ii) In the r ange T 0 < t ≤ T 1 , the numb er of p artitions λ of N with H t ( λ ) ≤ √ 6 N 4 π q t is ≪ N − 10 p ( N ) . Pr o of. Given an in teger t , and a parameter k w e wish to b ound the n um b er of partitions λ of N with H t ( λ ) ≤ k . F or any y ∈ (0 , 1) this is ≤ y − k X λ ⊢ N y H t ( λ ) ≤ y − k q − N X λ q | λ | y H t ( λ ) , whic h, b y Han’s form ula (11) and the asymptotic (9), is ≤ y − k q − N F ( q ) ∞ Y j =1 (1 − q tj ) t (1 − y j q tj ) t ≪ N 3 4 p ( N ) y − k ∞ Y j =1 (1 − q tj ) t (1 − y j q tj ) t . No w, y − k ∞ Y j =1 (1 − q tj ) t (1 − y j q tj ) t = y − k exp  t ∞ X j =1 log (1 − q tj ) (1 − y j q tj )  = y − k exp  t ∞ X j =1 ∞ X ℓ =1 ( y q t ) j ℓ − q tj ℓ ℓ  , and grouping terms according to n = j ℓ , w e conclude that the num ber of partitions of N with H t ( λ ) ≤ k is (15) ≪ N 3 4 p ( N ) y − k exp  − t ∞ X n =1 σ ( n ) n (1 − y n ) q tn  . ZER OS IN THE CHARA CTER T ABLE OF THE SYMMETRIC GROUP 7 T o prov e part (i), w e choose y = q t/ log N , and tak e k = N t − 1 (1 − 2 / log N ). Using (5) t wice (with x = √ 6 N / ( π t ) and √ 6 N / ( π t )(1 + 1 / log N ) − 1 b oth of which are ≫ (log N ) 4 in the range t ≤ T 0 ), the b ound in (15) is ≪ N 3 4 p ( N ) exp  π √ N √ 6 log N  1 − 2 log N  − t  π √ N √ 6 t 1 1 + log N − 1 log N + O ((log N ) − 2 )  . This simplifies to give the b ound ≪ N 3 4 p ( N ) exp  − π √ N √ 6(log N ) 2 + t log N + O  t (log N ) 2  , whic h is muc h smaller than N − 10 p ( N ) in the range t ≤ T 0 . In the range T 0 < t ≤ T 1 , w e c ho ose y = q t and k = √ 6 N q t / (4 π ) in (15). Note that y − k = exp( tq t / 4), and using σ ( n ) ≥ n and 1 − y n ≥ (1 − q t ) for all n ≥ 1, t ∞ X n =1 σ ( n ) n (1 − y n ) q tn ≥ t (1 − q t ) q t 1 − q t = tq t . Th us the b ound in (15) shows that the n um b er of desired partitions is ≪ N 3 4 p ( N ) exp( − 3 4 tq t ) ≪ N − 10 p ( N ) for T 0 < t ≤ T 1 . This completes our pro of. □ In the range t ≥ T 1 , w e also need an understanding of c t ( N ), the n um b er of t -core partitions of N , whic h is pro vided by the next prop osition. Much more is known ab out the num b er of t -cores c t ( N ), and for a thorough treatment see work of Tyler [12]. Prop osition 2. L et N b e lar ge, and r e c al l that q = exp( − π / √ 6 N ) . If t ≥ T 1 , then the numb er of p artitions λ of N with H t ( λ ) = 0 (in other wor ds, t -c or es λ ) is c t ( N ) = p ( N )  exp( − tq t ) + O ( N − 1 12 )  . Pr o of. Using the generating function (12), w e obtain c t ( N ) = 1 2 π Z π − π F ( q e iθ ) ∞ Y n =1 (1 − ( q e iθ ) tn ) t q − N e − iN θ dθ . No w for t ≥ T 1 , we hav e q t ≪ N − 1 2 and tq t ≪ log N , and therefore ∞ Y n =1 (1 − ( q e iθ ) tn ) t = exp  − tq t e itθ + O ( tq 2 t )  = exp  − tq t + O ( tq 2 t + tq t min(1 , t | θ | ))  . In view of (7), the contribution to c t ( N ) from the p ortion of the integral with | θ | ≥ N − 3 5 is muc h smaller than p ( N ) / N . Restricting no w to the p ortion | θ | ≤ N − 3 5 , w e find (since t 2 q t ≪ N 1 2 + ϵ for t > T 1 ) c t ( N ) = 1 2 π Z | θ |≤ N − 3 5 F ( q e iθ ) q − N e − iN θ exp  − tq t + O ( N − 1 12 )  dθ + O ( p ( N ) / N ) , and the result follows from (8) and (10). □ 8 SARAH PELUSE AND KANNAN SOUNDARARAJAN 4. Distribution of p ar ts Giv en a partition µ of N and a natural n umber t , recall that P t ( µ ) = 1 t X t | µ j µ j , so that t P t ( µ ) equals the sum o ver the parts in µ that are multiples of t . In suitable ranges of t , w e shall show that P t ( µ ) is suitably small for most partitions µ . Clearly P 1 ( µ ) = N for all partitions µ of N , and we may restrict atten tion to t ≥ 2. Much is known ab out the structure of a random partition (for example, see [1]), but our fo cus is on quick results that are sufficient for our purp oses. Prop osition 3. L et N b e lar ge and r e c al l that q = exp( − π / √ 6 N ) . Uniformly for al l 1 ≤ t ≤ N , the numb er of p artitions µ of N with at le ast one p art e qual to t is p ( N − t ) = q t p ( N )  1 + O  min  1 , t N 3 4  , and in p articular this is always ≪ q t p ( N ) . Pr o of. Adjoining a part t to any partition of N − t gives a bijective corresp ondence with partitions of N having at least one part equal to t . By the integral form ula p ( N − t ) = 1 2 π Z π − π F ( q e iθ ) q − N + t e − iN θ e itθ dθ = q t 2 π Z π − π F ( q e iθ ) q − N e − iN θ  1+ O (min(1 , t | θ | )  dθ . The main term ab ov e equals q t p ( N ) and the remainder terms may b e b ounded using (7) and (9). The prop osition follows. □ Recall the thresholds T 0 and T 1 defined in (13) ab ov e. Prop osition 4. L et N b e lar ge and r e c al l that q = exp( − π / √ 6 N ) . (i) In the r ange 2 ≤ t ≤ T 0 , the numb er of p artitions µ of N with P t ( µ ) > 9 N 10 t is ≪ N − 10 p ( N ) . (ii) In the r ange T 0 < t ≤ T 1 , the numb er of p artitions µ of N with P t ( µ ) > √ 6 N 4 π q t is ≪ N − 10 p ( N ) . (iii) In the r ange t ≥ T 1 , the numb er of p artitions µ of N with P t ( µ ) ≥ 2 is ≪ q 2 t p ( N ) . Pr o of. Let a t ( n ) denote the n umber of partitions of n in to parts that are not m ultiples of t . Separate the parts in a partition of N in to the parts that are m ultiples of t , and into the parts that are not multiples of t . The n um b er of partitions λ of N with P t ( µ ) = ℓ is clearly p ( ℓ ) a t ( N − tℓ ). Therefore, for any k , the n um b er of partitions µ with P t ( µ ) > k is (16) X ℓ>k p ( ℓ ) a t ( N − tℓ ) ≤ X ℓ>k p ( ℓ ) p ( N − tℓ ) , since a t ( n ) ≤ p ( n ) alw ays. ZER OS IN THE CHARA CTER T ABLE OF THE SYMMETRIC GROUP 9 In the range 2 ≤ t ≤ T 0 , w e use the b ound in (16) taking k = 9 N 10 t , together with the crude estimate p ( n ) ≪ exp(2 π √ n/ √ 6). A small calculation sho ws a b ound m uc h smaller than N − 10 p ( N ). A similar argumen t with k = √ 6 N 4 π q t w orks in the range T 0 ≤ t ≤ T 1 . Finally , the same argument giv es that for t ≥ T 1 , the n umber of partitions µ with P t ( µ ) ≥ 2 is (using Prop osition 3) ≪ X ℓ ≥ 2 p ( ℓ ) p ( N − ℓt ) ≪ p ( N ) X ℓ ≥ 2 p ( ℓ ) q ℓt ≪ q 2 t p ( N ) . This completes our pro of. □ 5. Proof of the main theorem W e will no w count the num ber of pairs ( λ, µ ) of partitions of N satisfying Stanley’s criterion (1). Equiv alen tly , these are the pairs ( λ, µ ) for whic h there exists a natural n umber t with P t ( µ ) > H t ( λ ). Since H 1 ( λ ) = P 1 ( µ ) = N , we ma y assume that t ≥ 2. There are sev eral differen t w a ys in whic h this could happ en, and we consider the following p ossibilities: (i) F or some 2 ≤ t ≤ T 1 w e ha v e P t ( µ ) > H t ( λ ). (ii) F or some t > T 1 one has P t ( µ ) ≥ 2. (iii) There exists T 1 < t ≤ T 2 with P t ( µ ) = 1 and H t ( λ ) = 0. (iv) The partition µ contains t w o parts t 1 < t 2 that are b oth larger than T 2 . (v) The partition µ has a unique part t > T 2 and λ is a t -core. An y pair ( λ, µ ) satisfying Stanley’s criterion m ust satisfy at least one of these fiv e condi- tions ab o v e. W e will sho w that the num ber of pairs ( λ, µ ) satisfying either of the conditions (i) or (ii) ab ov e is (17) ≪ N − 1 2 p ( N ) 2 , while the num ber of pairs ( λ, µ ) satisfying either condition (iii) or (iv) is (18) ≪ p ( N ) 2 (log log N ) 2 (log N ) 2 , and lastly the num ber of pairs satisfying condition (v) equals (19) p ( N ) 2  2 log N + O  log log N (log N ) 2  . Since every pair satisfying condition (v) corresp onds to a zero of type I, and zeros of type I I I that are not of type I I m ust satisfy either condition (i) or (ii), our main Theorem w ould follo w at once. W e begin with b ounding the pairs satisfying condition (i). Given an in teger t ≤ T 0 , if P t ( µ ) > H t ( λ ) then either H t ( λ ) ≤ ( N /t )(1 − 2 / log N ) or P t ( µ ) > 9 N / (10 t ). By Prop o- sitions 1 and 4, the n umber of suc h pairs ( λ, µ ) is ≪ p ( N ) 2 N − 10 . Summing this o v er all the p ossibilities for t , we conclude that there are at most ≪ N − 9 p ( N ) 2 pairs ( λ, µ ) with P t ( µ ) > H t ( λ ) for some 2 ≤ t ≤ T 0 . Similarly , Prop ositions 1 and 4 show that there are at 10 SARAH PELUSE AND KANNAN SOUNDARARAJAN most ≪ N − 9 p ( N ) 2 pairs ( λ, µ ) with P t ( µ ) > √ 6 N 4 π q t or H t ( λ ) ≤ √ 6 N 4 π q t for some T 0 ≤ t ≤ T 1 . W e thus conclude that the n umber of pairs satisfying condition (i) is m uch smaller than the b ound claimed in (17). W e mo ve no w to condition (ii). By part (iii) of Prop osition 4, the num b er of partitions µ with P t ( µ ) ≥ 2 for some t > T 1 is ≪ X t>T 1 q 2 t p ( N ) ≪ q 2 T 1 1 − q 2 p ( N ) ≪ √ N q 2 T 1 p ( N ) ≪ N − 1 2 p ( N ) . Th us there are ≪ N − 1 2 p ( N ) 2 pairs of partitions satisfying condition (ii), whic h is the bound in (17). No w consider condition (iii), and supp ose that t is in the range T 1 < t ≤ T 2 . By Prop osi- tions 2 and 3, the num b er of pairs ( λ, µ ) with P t ( µ ) = 1 and H t ( λ ) = 0 for some t in this range is ≪ p ( N ) 2 X T 1 t 1 >T 2 p ( N − t 1 − t 2 ) ≪ p ( N ) X t 2 >t 2 >T 2 q t 1 + t 2 ≪ p ( N )  q T 2 1 − q  2 ≪ p ( N ) (log log N ) 2 (log N ) 2 . There are thus at most ≪ p ( N ) 2 (log log N ) 2 / (log N ) 2 pairs satisfying condition (iv). Finally we are left with case (v). Here we m ust count partitions µ that hav e a unique part t > T 2 (so that P t ( µ ) = 1) and partitions λ that are t -core s. Since case (iv) rev eals that there are few partitions µ with tw o parts larger than T 2 , we may ignore the uniqueness condition, and fo cus simply on partitions µ containing a part t > T 2 and corresp onding t -cores λ . By Prop ositions 2 and 3 the num b er of such pairs is p ( N ) 2 X t>T 2  exp( − tq t ) + O ( N − 1 12 )  q t  1 + O  min  1 , t N 3 4  = p ( N ) 2  X t>T 2 q t exp( − tq t ) + O ( N − 1 12 )  . ZER OS IN THE CHARA CTER T ABLE OF THE SYMMETRIC GROUP 11 T o ev aluate the sum ab ov e, note that for t > T 2 q t exp( − tq t ) = Z t +1 t q x exp( − xq x ) dx + O  max t ≤ x ≤ t +1    ( q x exp( − xq x )) ′     = Z t +1 t q x exp( − xq x ) dx + O ( q t N − 1 2 ) , so that X t>T 2 q t exp( − tq t ) = Z ∞ T 2 q x exp( − xq x ) dx + O  X t>T 2 q t N − 1 2  = Z ∞ 0 q T 2 + y exp( − ( T 2 + y ) q T 2 + y ) dy + O ( N − 1 2 ) . No w for y ≥ 0 (using e − ξ = 1 + O ( ξ ) for ξ > 0 and that y q y ≤ 1 / log (1 /q ) for y > 0) exp( − ( T 2 + y ) q T 2 + y ) = exp( − T 2 q T 2 + y )  1 + O ( y q T 2 + y )  = exp( − T 2 q T 2 + y )  1 + O  q T 2 log(1 /q )  = exp( − T 2 q T 2 + y )  1 + O  log log N log N  , and Z ∞ 0 q T 2 + y exp( − T 2 q T 2 + y ) dy = 1 − exp( − T 2 q T 2 ) T 2 log(1 /q ) = 2 log N + O  log log N (log N ) 2  . Th us the count of pairs satisfying condition (v) ob eys the asymptotic (19), and the pro of of Theorem 1 is complete. References [1] B. F ristedt. The structure of random partitions of large integers. T r ans. Amer. Math. So c. , 337(2):703– 735, 1993. [2] G.-N. Han. The Nekraso v-Okounk o v ho ok length form ula: refinement, elementary pro of, extension and applications. Ann. Inst. F ourier (Gr enoble) , 60(1):1–29, 2010. [3] G. James and A. Kerb er. The r epr esentation the ory of the symmetric gr oup , volume 16 of Encyclop e dia of Mathematics and its Applic ations . 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Stanley . The stable b eha vior of some characters of SL( n, C ). Line ar and Multiline ar Algebr a , 16(1-4):3–27, 1984. [11] R. P . Stanley . Enumer ative c ombinatorics. Vol. 2 , volume 62 of Cambridge Studies in A dvanc e d Math- ematics . Cambridge Universit y Press, Cam bridge, 1999. With a foreword by Gian-Carlo Rota and app endix 1 by Sergey F omin. [12] M. Tyler. Asymptotics for t-core partitions and Stanton’s conjecture. A dvanc es in Mathematics , 489:110805, 2026. Dep ar tment of Ma thema tics, St anford University, St anford, CA, USA Email addr ess : speluse@stanford.edu Email addr ess : ksound@stanford.edu