Partial symmetries of iterated plethysms

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ “This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00026-023-00652-4” Ann. Comb. 27 (2023) 493–518 c 2023 The Author(s), under exclusive licence to Springer Nature Switzerland AG Published online May 29, 2023 https://doi.org/10.1007/s00026-023-00652-4 Annals of Combinatorics Partial Symmetries of Iterated Plethysms ´ Alvaro Guti´errez and Mercedes H. Rosas Abstract. This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous ωinvolution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with sλpreserves flip-symmetry, provided that λis a partition of two. Explicit formulas for the iterated plethysms s2◦sb◦saand sc◦s2◦sa,witha, b, and c≥2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients. Mathematics Subject Classification. 05E05, 05E18, 05A17. Keywords. Symmetric functions, Plethysm. 1. Introduction Let Vbe an ndimensional complex vector space. Each partition λof length at most nindexes an irreducible representation (unique up to isomorphism) of the complex general lineal group GL(V).The irreducible representation indexed by λcan be constructed as the evaluation of the Schur functor Sλon the vector space V. Therefore, it is denoted by Sλ[V]. The composition of representations provides us with a important and natural way of combining group representations, an operation referred to as the plethysm of representations. In the setting of the representation theory of the general lineal group, the plethysm of the irreducible representations indexed by μand νis defined by the composition of Schur functors Sμ[Sν[V]]. Further information can be found in Fulton and Harris’ book [1]. 494 ´ A. Guti´errez and M. H. Rosas Rational representations of the general linear group are completely reducible. This raises the question of decomposing the plethysm Sμ[Sν[V]] as a sum of irreducible representations. The plethystic coefficient aλ μ[ν]is defined as the multiplicity of Sλ[V]in Sμ[Sν[V]]. More generally, the iterated plethystic coefficient aλ μ1[μ2[...[μk]]] is defined as the multiplicity of Sλ[V]inSμ1[Sμ2[...[Sμk[V]]]]. The partitions indexing the iterated plethystic coefficient aλ μ1[μ2[...[μk]]] satisfy that |λ|= |μ1||μ2|···|μk|. The problem of understanding the plethystic coefficients is a notoriously hard problem [1–4] that has stumped many attempts to solve it. In this work, we report the occurrence of partial symmetries in certain iterated plethystic coefficients indexed by a specific type of partitions, hook+column partitions. First considered by Langley and Remmel [5] in 2004, a hook+column partition is a partition of the form (α, 2β,1γ). Langley and Remmel obtained a simple formula (stated in Theorem 2.14) for the plethystic coefficients aλ (b)[(a)], where λis a hook+column partition. Following them, we restrict our attention to iterated plethystic coefficients aλ μ1[μ2[...[μk]]] where λis a hook+column, and where each partition μiis either a row or a column partition. We derive closed formulas for the plethystic coefficients aλ (c)[(b)[(a)]] when either bor cis equal to 2, and λis a hook+column partition. These formulas (described in Theorems 2.15 and 2.16) allow us to uncover new and intriguing partial symmetries in the iterated plethystic coefficients that we illustrate in the following example. Example 1.1. Consider the coefficients aλ[ [ ]] with λhook+column partitions. The non-zero coefficients are: a[ [ ]] =1, a[ [ ]] =2, a[ [ ]] =3, a[ [ ]] =3, a[ [ ]] =2, a[ [ ]] =1, a[ [ ]] =1, a[ [ ]] =1. The symmetry demonstrated in this example can be characterized by a generalization of the transposition of diagrams where some (2 ×1) horizontal dominoes in λswap their placement (between the first two columns and the first row) without altering their orientation. We name the operation that describes this process the flip involution. It is a partial symmetry of S(2)[S(3)[S(2)[V]]] because it is only defined on the coefficients aλ[ [ ]] for λ a hook+column. This situation evokes the partial symmetry for the Littlewood–Richardson coefficients, originally conjectured to exist by Pelletier and Ressayre [6], and described explicitly and shown to hold by Grinberg [7]. The presence of symmetries often gives us a better grasp of these coefficients. They can be useful Partial Symmetries of Iterated Plethysms 495 in simplifying the number of cases that need to be addressed in both proofs and calculations involving them [8]. In Sect. 2, we provide an overview of the necessary background on symmetric functions, and present the following results. We introduce a partial involution, defined on hook+column partitions, in Definition 2.8, named the flip involution. Lemma 2.11 shows that this involution can be understood as a “transposition” of brick tilings of partitions. We then give closed expressions for the multiplicities of hook+column irreducible representations in S(c)[S(b)[S(a)[V]]], when either bor cis equal to 2, in Theorems 2.16 and 2.15.As a corollary, these are examples of flip-symmetric representations. In Theorem 2.19, we show that the Schur functors S(2) and S(1,1) preserve flip-symmetry. This result allows us to produce infinite families of flip-symmetric representations. In Sect. 3, we furnish proofs for all these results. In Sect. 4, we finish this article presenting an analysis of certain intriguing sequences constructed in a natural way from flip-symmetric iterated plethysm coefficients that appear to be both unimodal and asymptotically normal. 2. Background and Presentation of the Results Let Va complex vector space of dimension n. In the language of symmetric functions, the role of the irreducible representation Sλ[V]ofGL(V)isplayed by the Schur polynomial sλ(x1,x 2,...,x n).A representation Wof GL(V) will then corresponds to a Schur-positive symmetric function f, and the multiplicity of Sλ[V]inWis equal to the coefficient of sλin f. We follow Stanley [9] for the standard concepts and notations in the theory of symmetric functions, the main exception being that we write our partitions using the French convention [10]. A partition is a weakly decreasing sequence of natural numbers in which there are finitely many non-zero entries. The non-zero entries are called the parts. Two partitions are equal if they have the same parts. The weight of a partition is defined as the sum of its parts. We let Par(N) be the set of partitions of weight N. We identify partitions with their Young diagrams. The Young diagram of a partition λ=(λ1,λ 2,...) is the set {(c, r):0≤c≤λr}, whose elements we call cells. According to the French convention Young diagrams are drawn bottom-left justified, thus cells of the diagram are described by its Cartesian coordinates. The transpose λof λis the partition whose diagram is the image of the diagram of λunder the reflection (c, r)→ (r, c). Given partitions μand λ,the skew-partition μ/λ is the set of cells in μbut not in λ. We define the sum of two partitions λand μas the partition λ+μ=(λ1+μ1,λ 2+μ2,...). The union of two partitions is the sorting of its parts. Definition 2.1. We say a partition is a hook+column partition if it is of the form (α, 2β,1γ), that is, the sum of a hook partition and a column partition. Hook+column partitions were introduced by Langley and Remmel [5] along with hook+row partitions (α, β, 1γ), which are the union of a hook and a row partition. 496 ´ A. Guti´errez and M. H. Rosas Warning 2.2. When we write λ=(λ1,2m2(λ),1m1(λ)) for a hook+column partition, we use the convention that λ1can be equal to either one or two. We write (15)as(1,20,14), and (22,1) as (2,21,11).Thus, m1(λ)andm2(λ)are not always the multiplicities of 1 and 2 in λ. The algebra of symmetric functions Λ is defined as the algebra Q[p1,p 2,...] generated by a set of variables that play the role of the power sum symmetric functions. Note that these are not defined in terms of another set of variables. However, it will sometimes be useful to identify an element fof Λ with a formal power series. For this, we let X=x1+x2+··· be an alphabet. (The xi’s are variables.) Then, we identify f∈Λ with its image f[X] under the algebra morphism that maps pkto xk 1+xk 2+···. We write f[X]=f(x1,x 2,...)and say that it is the evaluation of fin X. In particular, we identify p1with X=x1+x2+···. The notion of plethysm of symmetric functions comes from that of composition of representations. Consider the action of a Schur functor Sλon a diagonal endomorphism whose eigenvalues are variables. Its trace is then the Schur polynomial sλ. Therefore to compute the plethysm of two symmetric functions one has to substitute the monomials appearing in the first symmetric function into the second one. More precisely, let fand gbe elements of Λ. If g[X]isa sum of monic monomials, g[X]=g1+g2+··· then f◦g[X]=f(g1,g 2,...). Example 2.3. Let f[X]=2p2[X], then, since 2p2[X]=2x2 1+2x2 2+··· = x2 1+x2 1+x2 2+x2 2+···,wehavethat pn[2p2[X]]=pn(x2 1,x 2 1,x 2 2,x 2 2,...)=2p2n[X]. Warning 2.4. One needs to be careful using plethystic notation. In general, evaluating on alphabet cX is not equivalent to evaluating on the alphabet cx1+cx2+···. We denote the first by f[cX] and the latter by f[tX]|t=c.In particular, −pk[X]=pk[−X]=pk[tX]|t=−1=(−1)kpk[X]. The plethysm of symmetric functions can be defined axiomatically. Definition 2.5. (Plethysm of symmetric functions) The plethysm of symmetric functions is the operation ◦:Λ×Λ→Λ verifying 1. pn◦pm=pnm for all n, m ∈N0. 2. For any f∈Λ, the map g→ g◦fis a Z-algebra homomorphism on Λ. 3. For any f∈Λ, the equality pn◦f=f◦pnholds. The core tools of this work come from plethystic calculus, namely, from the operation of evaluation in sums and differences of alphabets. The following lemma is a consequence of the well known expansion of sλ◦(f±g) found in [10]. Let cλ μ,ν denote the Littlewood–Richardson coefficient indexed by partitions μ, ν and λ. That is, cλ μ,ν is the coefficient of sλin the product sμ·sν. Lemma 2.6. Let Xand Ybe two alphabets and let λbe a partition. Then: 1. sλ[−X]=(−1)|λ|sλ[X]. 2. sλ[X+Y]=μ⊂λsμ[X]·sλ/μ[Y]=μ,ν cλ μ,ν sμ[X]·sν[Y]. Partial Symmetries of Iterated Plethysms 497 4 2 3 3 1 1 1 3 −1 −4−3 −5−4 −5−4−2 3 −1 −2 2 3 −2 1 1 3 Figure 1. Three valid SSYT with positive and/or negative letters 3. sλ[X−Y]=μ⊂λ(−1)|λ/μ|sμ[X]·s(λ/μ)[Y] =μ,ν(−1)|ν|cλ μ,ν sμ[X]·sν[Y]. Note 2.7. Let Xand Ybe two alphabets. Then, Lemma 2.6 says that sλ[X−Y] is the generating function of the tableaux Ton positive letters from Xand negative letters from −Yobeying the semistandard rules for the positive entries and a similar, but opposite rule for the negative ones: negative entries should be weakly increasing across the columns when reading from from bottom to top, and strictly increasing across the rows, when reading from left to right. See Fig. 1. Definition 2.8. Fix a non-negative number γ, and let fbe a Schur positive and homogeneous symmetric function of degree n. Define bβ f,γ as the coefficient of s(n−2β−γ,2β,1γ)in f. Then, The hook+column sequence Σ(f,γ) is defined as the sequence Σ(f,γ)=(bβ f,γ)β≥0. We refer to a sequence as symmetric if its non-zero entries form a symmetric sequence. Example 2.9. Tables 1and 2give examples of hook+column sequences. The data clearly indicates that these are symmetric sequences. Some further properties of these sequences will be discussed in the final section of this work. We describe the symmetry present in these sequences using the flip involution, a partial involution on partitions that we proceed to define. Definition 2.10. Let λbe a hook+column partition, and let r≥2 be an integer. Assume furthermore that one can write λ1−r−γ=2δfor some non-negative integer δ.Theflip involution with offset r, also called r-flip, is defined as follows: λ=(r+2δ+γ,2β,1γ)→ Flip(r;λ)=(r+2β+γ,2δ,1γ). Note that Flip(r;−) is clearly an involution on its domain. See Fig. 2. The following lemma translates the r-flip involution to a tiled transposition of Young diagrams. We say a partition is tiled if we have a collection of non-overlapping rectangles (tiles) of shape (h×w) for some h, w ∈Ncovering its Young diagram. A tiling is a brick tiling if every tile is of height 1. 498 ´ A. Guti´errez and M. H. Rosas Table 1. The hook+column sequence of sc◦sb◦s2and γ=0,forb=3andc=1,...,6 Function, fγHook+column sequence, Σ(f,γ) s1◦s3◦s20 (1, 1, 1) s2◦s3◦s20 (1, 2, 3, 3, 2, 1) s3◦s3◦s20 (1, 2, 5, 7, 8, 7, 5, 2, 1) s4◦s3◦s20 (1, 2, 5, 10, 15, 18, 18, 15, 10, 5, 2, 1) s5◦s3◦s20 (1, 2, 5, 10, 15, 19, 28, 36, 38, 36, 28, 19, 10, 5, 2, 1) s6◦s3◦s20 (1, 2, 5, 10, 19, 33, 49, 63, 72, 72, 63, 49, 33, 19, 10, 5, 2, 1) Partial Symmetries of Iterated Plethysms 499 Table 2. The hook+column sequence of sc◦sb◦s2and γ=0,forb=4andc=1,...,6 Function, fγHook+column sequence, Σ(f,γ) s1◦s4◦s20 (1, 1, 1, 1) s2◦s4◦s20 (1, 2, 3, 4, 4, 3, 2, 1) s3◦s4◦s20 (1, 2, 5, 8, 11, 13, 13, 11, 8, 5, 2, 1) s4◦s4◦s20 (1, 2, 5, 11, 18, 26, 34, 38, 38, 34, 26, 18, 11, 5, 2, 1) s5◦s4◦s20 (1, 2, 5, 11, 22, 36, 55, 74, 90, 100, 100, 90, 74, 55, 36, ...) s6◦s4◦s20 (1, 2, 5, 11, 22, 41, 68, 103, 144, 184, 217, 236, 236, 217, 184, ...) 500 ´ A. Guti´errez and M. H. Rosas (6,23,1) = = (10,2,1) offset offset 3-flip →−−− Figure 2. The three-flip of (6,23,1), is Flip(3; (6,23,1)) = (10,2,1) ( 6,23,1 ) → 1 2 2 2 3 2 1 → 1 2 3 2 2 2 1 → (10,2,1) Figure 3. The tiled transpose of this tiling of (6,23,1) is Flip(3; (6,23,1)) = (10,2,1) A brick tiling of a Young diagram corresponds to a tableau in the following manner: a row tiled into (1 ×t1),(1 ×t2),...,(1 ×tk) can be collapsed to the row-tableau t1t2···tk; now the diagram can be collapsed row by row into a tableau Tλ. (The shape of this tableau need not be a partition, but a composition.) If the shape of Tλis a partition, we can define the tiled-transpose of λas the (brick tiled) partition μwhose tableau Tμis the transpose of Tλ. Lemma 2.11. Let λ=(λ1,2β,1γ)be a hook+column partition and let r≥2be an integer. Assume furthermore that one can write λ1−r−γ=2δfor some non-negative integer δ. Consider the following brick tiling of λ. 1. The first row is made up of one (1 ×r)-tile, followed by δtiles of shape (1 ×2),andγtiles of shape (1 ×1). 2. Each of the remaining rows forms a tile. Then the flip involution with offset rcorresponds to tiled transposition. See Fig. 3. Given a symmetric function fand a partition μ, we denote the coefficient of sμin the expansion of fin the Schur basis by [μ]f. We define the support of fas the set of partitions appearing in the decomposition of fin the Schur basis, supp(f)={μ:[μ]f=0}. Definition 2.12. A symmetric function fis flip-symmetric with offset rif for all hook+column μin supp(f), 1. Flip(r;−) is defined on μ,and 2. [ν]f= [Flip(r;μ)]f. Moreover, fis flip-symmetric if its flip-symmetric with some offset r. In particular, hook+column sequences of flip-symmetric functions are symmetric sequences. However, as the following example illustrates, not all Partial Symmetries of Iterated Plethysms 507 j i i=j j=b−1 i+j=2b−k−1 i+j=2b−k Figure 5. We let b=5andk= 5. The polytope Δ is shaded in blue. Each black dot represents a valid pair. On the right, we illustrate the result of the described projection Consequently, the hook+column partitions that will appear in the sum are those whose first row is in the set {2,4,6,...,4b}. We now ask how many times each one appear. Take b∈N≥2and k∈{1,2,...,2b}. We ask how many integer pairs (i, j) there are in the polytope Δ:= {0≤i≤j<b}, which are solutions to either of these two equations: 4b−2(i+j)=2k, 4b−2(i+j+1)=2k, or, consequently, i+j=2b−k, i+j=2b−k−1.(2) Refer to Fig. 5for a graphical representation. To help count the solutions, we will project them orthogonally from one of the lines to the other one, in such a way that all the projections remain inside Δ. More precisely, if k≥bthen project onto {i+j=2b−k},anif k<bto the other line. One can easily see now what the coefficients are going to look like. Noting that the biggest line is counted twice (because we change the projection mid way), results in the desired integer sequence for γ=0. Let us now bring back Equation (1) and restrict to the γ= 0 part of the hook+column sequence. Using [5, Thm. 4.8 (3)] again, we obtain (s2◦sb◦s2)|γ=0 h+c =⎛ ⎝ i<j<b s(2b−2i,2i)·s(2b−2j,2j)⎞ ⎠ γ=0 h+c . These products give rise, using Lemma 3.4, to a new polytope {0≤i<j<b} together with the equation i+j=2b−k. A similar argument gives the desired sequence. This completes the proof of Theorem 2.15. 3.3. An Explicit Formula for sc◦s2◦saon Hook+Columns This section is dedicated to the proof of Theorem 2.16. Fixing γ=0,we will show that the first cterms of the hook+column sequence are given by the formula n2+n+2 2(OEIS A000124 [12]) that have, as generating function z (1−z)3+1 1−z[16]. As before, it will be enough to show it for a=2.The techniques used in this section have been used by the second author in [11]to study the Kronecker coefficients, and by Langley and Remmel [5]. 508 ´ A. Guti´errez and M. H. Rosas ? . . . ? ? −2−1 . . . . . . −2−1 −2−1 1 ··· 1 Figure 6. Our only choices when filling a hook+column with 1, −1and−2 are represented by a question mark We begin by explicitly computing the evaluation of a Schur function indexed by a hook+column on the alphabet 1 −x−y. Lemma 3.8. Let λ=(α, 2β,1γ)and α≥2.Then, sλ[1 −x−y]=(−1)γ(xy)β(1 −x)(1 −y)xγ+1 −yγ+1 x−y. In particular, the formula is independent of α. Proof. Recall Note 2.7. We construct all tableaux of shape λwith three letters (fixing an order, let 1 be the letter for the variable 1, −1for−x,and−2for −y) and record the weight of each resulting tableau. Notice that we have very little freedom when filling a hook+column with only these three letters. Our only choices are in the last entries in the first two columns (see Fig. 6). We can have 0, 1 or 2 entries equal to 1 in these cells. The rest can be filled with various quantities of −1s and −2s, resulting into weights (xy −x−y+1)  i+j=γ (−1)γxiyj=(−1)γ(1 −x)(1 −y)xγ+1 −yγ+1 x−y. As there are βinstances of −2−1, the desired expression arises.  Further inspection of Lemma 3.8 reveals that the restriction to γ=0 equates to restriction to monomials which are also monomials in the variable (xy). We write this as follows: (sc◦s2◦s2)|γ=0 h+c [1 −x−y] (1 −x)(1 −y)=sc◦s2◦s2[1 −x−y] (1 −x)(1 −y)(xy) .(3) The numerator sc◦s2◦s2[1 −x−y] can be rewritten, using the equality sc=λcz−1 λpλand the properties of plethysm, as sc◦s2◦s2[1 −x−y]=sc◦(s4+s2,2)[1 −x−y](by2.14) =sc[(1 + xy)(1 −x)(1 −y)] (by 3.8) Partial Symmetries of Iterated Plethysms 509 = λc pλ[(1 + xy)(1 −x)(1 −y)] zλ = λc i1+(xy)λipλ[(1 −x)(1 −y)] zλ = k≥0 (xy)k λc χk(λ)pλ zλ [(1 −x)(1 −y)], where χk(λ)=⎧ ⎪ ⎨ ⎪ ⎩ μkm1(λ) m1(μ)···mk(λ) mk(μ)for 0 ≤k≤c 2, χc−k(λ)for c 2<k≤c, 0forc<k. Lemma 3.9. We have  λc χk(λ)pλ zλ=hc−k,k for 0≤k≤c 2. Proof. We compute hc−k,k =sc−ksk= μc−k pμ zμ νk pν zν = μc−k νk pμ∪ν zμzν = λc μk pλ zλmi(λ)! mi(μ)!(mi(λ)−mi(μ))! = λc χk(λ)pλ zλ .  Again by Note 2.7, we get the following lemma. Lemma 3.10. For n≥2, the have the following equality: sn[1 −x−y+xy]=(1−x)(1 −y)1−(xy)n−1 1−xy . Proof. A row tableau of size ncan have at most one entry equal to −1, and at most one entry equal to −2. The remaining cells n−2 can have and any number 0 ≤k≤nof entries equal to 1 and n−k−2 entries equal to 2.  Proof of Thm. 2.16 With these lemmas, expression (3) now becomes c k≥0(xy)khc−khk[(1 −x)(1 −y)] (1 −x)(1 −y)(xy) (4) 3.10 = 2c  k≥0 (xy)k+(1 −x)(1 −y) (1 −xy)2 c−1  k≥1 (xy)k−(xy)c−(xy)2k+(xy)c+k(xy) =1−(xy)2c+1 1−xy +1+xy (1 −xy)2 (xy)−(xy)c 1−xy −(n−1)(xy)c −(xy)2−(xy)2c 1−(xy)2+(xy)c−(xy)2c 1−xy !.(5) 510 ´ A. Guti´errez and M. H. Rosas Let us consider only the terms which affect the first centries of the hook+column sequence. We obtain 1 1−xy +1+xy (1 −xy)2 xy 1−xy −(xy)2 1−(xy)2!=1 1−xy +xy (1 −xy)3, which is precisely the generating function for OEIS A000124, as announced. So far, we have shown that sc◦s2◦s2yields, for γ= 0, a hook+column sequence starting with (1,2,4,...,T c+ 1), where Tcis the cth triangular number. Furthermore, as it is apparent from (4), the coefficient of (xy)kcoincides with the coefficient of (xy)c−k, proving that, in fact, the hook+column sequence of sc◦s2◦s2for γ= 0 is equal to (1,2,4,...,T c+1,T c+1,...,4,2,1,0,...). In other words, (sc◦s2◦sa)|γ=0 h+c is equal to 2c−1  k=0 min k2+k+2 2,(2c−1−k)2+(2c−1−k)+2 2·s(2ac−2k,2k). It remains to show that (sc◦s2◦sa)|γ=0 h+c is equal to 2c−3  k=1 min (k+1) 2 4,(2c−1−k)2 4·s(2ac−2k−1,2k,1). We only sketch the proof, since the computations are similar. We begin by considering the expression (5). But instead of restricting to monomials which are symmetric in xand y, we consider the remaining monomials. We obtain this way the generating function for the coefficients in for Σ(sc◦s2◦sa,1). This time, the starting sequence is an offset of OEIS A002620 [12], whose general term is (n+1)2 4and which has a generating function z (1+z)(1−z)3[16]. This is shown similarly. This completes the proof of Theorem 2.16. 3.4. Symmetry of Hook+Column Sequences This section is dedicated to the proof of our main result (Theorem 2.19). Express s2as 1 2(p1,1+p2)ands1,1as 1 2(p1,1−p2). Using the results and notations of Sect. 3.1, for each hook+column λwe can write 2[λ](s2◦f)=#D(λ;p1,1◦f)+sgn 2(λ)#D(λ;p2◦f), 2[λ](s1,1◦f)=#D(λ;p1,1◦f)−sgn2(λ)#D(λ;p2◦f). Our aim is to show that if fis flip-symmetric with offset r, then [λ](s2◦f) equals [Flip(2r−2; λ)] (s2◦f). Let us denote R=2r−2 and let λR:=Flip(R;λ). Note from Table 3that the sign function is invariant under the flip involution. Hence, proving #D(λ;p2◦f)=#D(λR;p2◦f) and #D(λ;p1,1◦f)= #D(λR;p1,1◦f) will suffice to show the theorem. We begin with well-definedness. Note that for any r≥2, we have R≥2. Fix λ=(λ1,2β,1γ)∈supp(s2◦f). We claim that the R-flip is well defined on λ. That is, we can find δ≥0 such that λ1=R+2δ+γ. Indeed, the parity of 2n=λ1+2β+γimplies λ1−γ≡R≡0 mod 2. Moreover, suppose μ∈ D(λ;p2◦f), λ1is even and 2μ1=λ1. Then, r≤μ1−m1(μ)andβ≥2m2(μ) Partial Symmetries of Iterated Plethysms 511 imply R≤λ1−γ, as desired. In any other case (e.g., μ∈D0(λ;p1,1◦f)), similar computations yield similar results. The following four facts complete the proof of the theorem: •If λ1is odd, then #D(λ;p2◦f)=#D(λR;p2◦f) by Lemma 3.11. •If λ1is even, then #D0(λ;p2◦f)=#D2(λR;p2◦f) by Lemma 3.12. Relabelling λfor λRgives #D0(λR;p2◦f)=#D2(λ;p2◦f). •By Lemma 3.13,#D0(λ;p1,1◦f)=#D2(λR;p1,1◦f), and relabelling λ for λRgives #D0(λR;p1,1◦f)=#D2(λ;p1,1◦f). •Finally, #D1(λ;p1,1◦f)=#D1(λR;p1,1◦f) by Lemma 3.14. The non-negativity of the resulting coefficients holds from Lemma 3.1 and from the fact that Schur positivity is preserved under plethysm by s2. Lemma 3.11. Under the hypotheses of this section, if λ1is odd then #D(λ;p2◦ f)=#D(λR;p2◦f). Proof. If λ1=R+2δ+γis odd, then so is λR 1=R+2β+γ. The multiset D(λ;p2◦f) only has one element, μ.Wehave μ= r+δ+γ−1 2,2β 2,1γ−1 2!r−flip −−−−→ r+β+γ−1 2,2δ 2,1γ−1 2!=: μr. Note that μris the only partition appearing in D(λR;p2◦f). By hypothesis, these two partitions appear with the same multiplicity [μ]f= [Flip(r;μ)] fin their corresponding multisets. This completes the proof.  The remaining lemmas will follow in the same spirit as the previous one. Lemma 3.12. Under the hypotheses of this section, if λ1is even then #D0(λ;p2◦ f)=#D2(λR;p2◦f). Proof. Explicitly, we can write D0(λ;p2◦f)andD2(λR;p2◦f) (up to multiplicities) as follows, r+δ+γ 2−1,2m2,1β+γ 2−2m2:m2≤β 2 and r+β+γ 2,2m 2,1•:m 2≤δ−1 2. Applying the r-flip on every element of D0(λ;p2◦f) yields Flip(r;D0(λ;p2◦f)) = r+β+γ 2,2m 2,1β+γ 2−2m2:m2≤β 2, where m 2=1 2(δ+2m2−β−1). We aim to identify Flip(r;D0(λ;p2◦f)) with D2(λR;p2◦f). The only thing that remains to show is that m2≤β 2if and only if m 2≤δ−1 2.Fromthe expression of m 2, the inequality m 2≤δ−1 2simplifies to δ−1+2m2−β 2≤δ−1 2, and it is now clear that this is equivalent to the inequality m2≤β 2. Lemma 3.13. Under the hypotheses of this section, we get #D0(λ;p1,1◦f)= #D2(λR;p1,1◦f). 512 ´ A. Guti´errez and M. H. Rosas Proof. Let μrand νrbe the r-flip of some (μ, ν)∈D0(λ;p1,1◦f). Note that |μ|+|ν|=|λ|and μ1+ν1=λ1imply 2m2(μ)+m1(μ)+2m2(ν)+m1(ν)=2β+γ. (6) Adding R=2r−2 to both sides of the equation, we conclude μ1+ν1=λ1if and only if μr 1+νr 1=λR 1+2. Again from Equation (6), we get m2≤βif and only if m1(μ)+m1(ν)≥γ. Knowing that μ1+ν1=λ1, we write r+2m2(μr)+m1(μ)+r+2m2(νr)+m1(ν)=R+2δ+γ =2r−2+2δ+γ. And so, m2≤βif and only if m 2+1≤δ, where m 2:=m2(μr)+m2(νr). In a similar fashion, one can show β≤m2+m1if and only if δ≤m 2+m1+1. Summing up, we have proved that a pair (μ, ν)isinD0(λ;p1,1◦f)ifand only if (μr,νr)isinD2(λR;p1,1◦f), and thus #D0(λ;p1,1◦f)=#D2(λR;p1,1◦ f) by the flip-symmetry hypothesis on f. Lemma 3.14. Under the hypotheses of this section, we get #D1(λ;p1,1◦f)= #D1(λR;p1,1◦f). Proof. The proof is similar to that of Lemma 3.13. Following in each step the case in which the equalities are attained, we also get χβ (μ,ν)=χδ (μr,νr). 4. Final Comments Theorems 2.15,2.16,and2.19 imply that the iterated plethysm f=s2◦s2◦ ···◦s2◦sc◦sb◦sais flip-symmetric when either bor cis equal to 2. Therefore, the hook+column sequence Σ(f,γ) is symmetric for each non-negative integer γ. However, sequences like this one appear to have stronger properties. Based on our data, we put forth some questions regarding the structural behavior of hook+column sequences arising from iterated plethysms. 1. Our first question is a very natural one. Are the hook+column sequences of the form Σ(sn1◦sn2◦···◦snk,γ) symmetric for all n1,...,n kand all γ? 2. Our algebraic definition of the flip involution makes sense even if the offset is r= 0 or 1. Our data suggests that if fis flip symmetric with offset 0or1,thenΣ(s2◦f,γ)andΣ(s1,1◦f,γ) are also symmetric sequences. But s2◦fand s1,1◦fare not flip-symmetric sequences according to our definition. See for instance Example 2.13. Based on our analysis of the data, we infer that there is a wider partial symmetry that has yet to be revealed. A more general description of the flip-symmetry and the flip involution is expected to exist. 3. A finite sequence (ai)i=1,...,n is said to be unimodal if there exists a k such that a0≤a1≤···≤ak−1≤ak≥ak+1 ≥···≥an−1≥an. Partial Symmetries of Iterated Plethysms 513 Based on the available data, we ask: Are the hook+column sequences Σ(sn1◦sn2◦···◦snk,γ) unimodal for all n1,...,n kand all γ? Another way of rephrasing this question is reminiscent of a celebrated result, the unimodality of the q-binomial coefficients. This result can be reduced to deciding whether the following polynomial is unimodal: (sa◦sb)[1 + q]=sa[(b+1) q]= a+b a!q = m bmqm, where (n)qis the q-analog of nand n kqis the q-binomial coefficient. On the other hand, our problem can be reduced to studying whether sn1◦sn2◦···◦snk[1 −x−y]·(x−y) (−1)γ(1 −x)(1 −y)(xγ+1yγ+1)= β aβ(xy)β is a unimodal polynomial in the variable (xy) for any γ. 4. A positive finite sequence (ai)i=1,...,n is said to be log-concave if a2 i≥ ai−1ai+1 for all i=2,...,n−1. It is well known that log-concavity implies unimodality. Recently, the log-concavity of many combinatorial sequences has been established thanks to the development of several breakthrough methods [17–20]. In some of these works, the sequences shown to be log-concave are not the classical combinatorial sequences, but a renormalization of them [21].The hook+column sequence Σ(s◦5 2,γ) appearing in Fig. 7gives us an example of a hook+column sequence that is not logconcave. However, is there a sensible renormalization of hook+column sequences arising from plethysm that renders them log-concave? 5. Asymptotic normality is another structural phenomenon commonly found in combinatorial sequences. Experimental evidence suggests that the hook +column sequences of (s2)◦k:=s2◦ktimes ... ◦s2for any fixed γare asymptotically normal when ktends to infinity, as Fig. 7illustrates. Moreover, aχ2normality test returns a p-value of 1 or almost 1 for every sequence coming from s◦k 2,k=2,...,5. These huge p-values1seem to indicate that the Gaussian curve perfectly fits our sequences, even for small values of k. Is the hook+column sequence Σ(s◦k 2,γ) asymptotically normal for each fixed γ? (In the sense that its relative sums approach a Gaussian curve when ktends to infinity.) (See [22–24] for more details in asymptotic normality of combinatorial integer sequences.) 6. We have used SageMath [25] to compute data supporting these questions. For instance, the hook+column sequences of the family s◦k 1,1also appear to be asymptotically normal. Moreover, our data for fabc:=sc◦sb◦sa suggest that the limiting hook+column sequences of fabc when both b and ctend to infinity is asymptotically normal (see Fig. 8). 7. In Theorems 2.15 and 2.16, we gave explicit descriptions of the hook+ column sequences Σ(fabc,γ), whenever bor care equal to 2. Our data 1We adopt the usual convention of rejecting the null hypothesis if the p-value is smaller than 0.05. 514 ´ A. Guti´errez and M. H. Rosas Function, f γ Hook+column sequence, Σ(f, γ) s◦2 20 (1, 1) s◦3 20 (1, 2, 2, 1) s◦4 20 (1, 3, 8, 13, 13, 8, 3, 1) s◦5 20 (1, 4, 20, 72, 205, 446, 756, 986, 986, 756, 446, 205, 72, 20, 4, 1) Figure 7. On top, a table showing the hook+column sequence of s◦k 2for γ=0andk=2,3,4,5. Below, plots of the aforementioned sequences with the x-axis being β,and represented as the normalized histogram whose frequencies read Σ(s◦k 2,0). They appear overlaid with Gaussian curves of adjusted mean and variance allows us to make reasonable guesses about what the hook+column sequences of fabc approach for other values of band c. (Recall that, by Lemma 3.7, the value of adoes not affect the non-vanishing part of the sequence.) The hook+column sequences of f23c=sc◦s3◦s2and γ=0 up to c= 6 are shown in Table 1. Unlike the sequences in Theorems 2.15 and 2.16, each consecutive sequence is not simply a longer version of the previous ones. However, they seem to stabilize. Is their stable limit sequence (1, 2, 5, 10, 19, 33, 57, 92, 147, ...),thenumberofpartitions with two kinds of 1 s, 2 s, and 3 s? (OEIS A000098 [12].)The hook+column sequences of f24c=sc◦s4◦s2and γ=0uptoc= 6 are shown in Table 2. Again, the coefficients seem to stabilize. Is their stable limit sequence (1, 2, 5, 11, 22, 42, 77, 135, ...), the number of partitions of 2n?(OEIS A058696 [12].) More generally, are the hook+column sequences Σ(fabc,γ) counting partitions of a number with some restriction on the allowable parts, for all b, c and γ? The literature already contains instances of sequences similar to the last two, in the context of exploring structural constants of symmetric functions, including Kronecker coefficients [26]. Partial Symmetries of Iterated Plethysms 515 Figure 8. From left to right, the histogram plots for the hook+column sequences associated to s9◦s2◦s2,s6◦s3◦s2, s5◦s4◦s2and s4◦s5◦s2, with γ= 0, and where the x-axis represents β. They appear overlaid with Gaussian curves of adjusted mean and variance Acknowledgements The authors express their appreciation to Adri`a Lillo, Emmanuel Briand, and Laura Colmenarejo for their insightful comments and engaging discussions. They also acknowledge the outstanding efforts of the two anonymous referees in thoroughly reviewing the preliminary version of their work. Author contributions All authors contributed equally to this manuscript. Funding MR was partially supported by MTM2016-75024-P and FEDER, PID2020-117843GB-I00, and Proyectos I+D+i FEDER Andaluc´ıa US-1262169. AG was partially supported by Junta de Andaluc´ıa FQM-333. Availability of Data and Materials Not applicable. Declarations Conflicts of Interest On behalf of all authors, the corresponding author states that there is no conflict of interest. Ethics Approval Not applicable. Code Availability Not applicable. Consent for Publication On behalf of all authors, the corresponding author gives consent for publication. Consent to Participate Not applicable. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of 516 ´ A. Guti´errez and M. H. Rosas this article is solely governed by the terms of such publishing agreement and applicable law. References [1] Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer-Verlag (1991). https://doi.org/10.1007/978-1-4612-0979-9 [2] Colmenarejo, L.: Stability properties of the plethysm: A combinatorial approach. Discrete Mathematics 340(8), 2020–2032 (2017). https://doi.org/10.1016/j.disc. 2016.10.009 [3] Fischer, N., Ikenmeyer, C.: The computational complexity of plethysm coefficients. Computational Complexity 29(2), 8 (2020). https://doi.org/10.1007/ s00037-020-00198-4 [4] Kahle, T., Michalek, M.: Plethysm and lattice point counting. Found. Comput. 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