Stanley 1980 weyl groups hard lefschetz theorem sperner

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corollary_5_1: Stanley's 1980 bound: if a set A of distinct reals has nu negative elements, zeta zeros and pi positive elements, then at most the sum of the k middle coefficients of 2^zeta (1+q)...(1+q^nu) (1+q)...(1+q^pi) subsets of A have element sums taking at most k values, with equality for the nonzero integers from -nu to pi together with 0 when zeta is 1; for positive sets and k equal to 1 the exact maximum of equal subset sums, attained by 1, ..., n.

corollary_5_3: Stanley's 1980 theorem that for a set of n distinct real numbers the number of subsets whose element sums take at most k values is at most the sum of the k middle coefficients of 2(1+q)...(1+q^nu)(1+q)...(1+q^pi), nu the integer part of (n-1)/2 and pi that of n/2, attained by the integers from -nu to pi; for k equal to 1 and n odd, the Erdős-Moser conjecture on the set maximizing the number of equal subset sums.

theorem_2_4: Stanley's main theorem of 1980: if a nonsingular irreducible complex projective variety X of complex dimension n has a cellular decomposition, then the poset Q^X of its cells, ordered by inclusion in closures, is graded of rank n, rank-symmetric, rank-unimodal and has the k-Sperner property for every k.

theorem_3_1: Stanley's 1980 theorem that for a Coxeter system (W, S) with W a Weyl group and any J contained in S, the poset W^J of minimal-length representatives of the cosets of W_J, under the Bruhat order, is rank-symmetric, rank-unimodal and has the k-Sperner property for every k.

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Stanley, Richard P., Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebraic Discrete Methods (1980), 168-184. The journal record is SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168--184, DOI 10.1137/0601021 (Crossref); received by the editors June 1, 1979 (p. 168).

Stanley shows that partially ordered sets Q^X arising from cellular decompositions of nonsingular irreducible complex projective varieties are graded, rank-symmetric, rank-unimodal and have the k-Sperner property for all k (Theorem 2.4), so the largest union of k antichains is the sum of the k largest rank sizes. Lemma 1.1 shows that a finite graded rank-symmetric poset of rank n is rank-unimodal with property S exactly when it has property T, and exactly when there are order-raising linear maps V_i -> V_{i+1} whose composites V_i -> V_{n-i} are invertible for i <= n/2; Theorem 2.1 identifies the cohomology basis coming from a cellular decomposition, and the hard Lefschetz theorem supplies the required invertibility. Applied to X = G/P for G a complex semisimple algebraic group and P parabolic, Q^X becomes the Bruhat order on a quotient of the Weyl group. Taking for P a certain maximal parabolic subgroup of G = SO(2n+1), Stanley deduces the following conjecture of Erdos and Moser: if S is a set of 2l+1 distinct real numbers and T_1,...,T_k are subsets of S whose element sums are all equal, then k is at most the middle coefficient of 2(1+q)^2(1+q^2)^2 ... (1+q^l)^2, and this bound is best possible. It bears on the first question of Erdos problem 362, which asks whether at most a constant times 2^N/N^{3/2} subsets of an N-element set of positive integers can share a sum: Corollaries 5.1 and 5.3 below give exact maxima, and the paper states no asymptotic order for them.

Source: https://math.mit.edu/~rstan/pubs/.

The edition read and the Section 5 corollaries. The copy read for this card is the seventeen-page PDF that the publications page above links as its item 42 (https://math.mit.edu/~rstan/pubs/pubfiles/42.pdf, byte-identical on 2026-10-07 to the copy read), a 600-dpi scan of the printed article with a text layer; PDF p. $n$ is printed p. $167+n$. The statement in the abstract (p. 168) is the case $n$ odd, $k=1$ of the general result of Section 5. Corollary 5.1 (p. 178): for a set $A$ of distinct real numbers with $\nu$ negative elements, $\zeta$ zeros ($\zeta=0$ or $1$) and $\pi$ positive elements, and subsets $B_1,\ldots,B_r$ of $A$ whose element sums take at most $k$ distinct values, $r$ is at most the sum of the $k$ middle coefficients of $G_{\nu\zeta\pi}(q)=2^\zeta(1+q)(1+q^2)\cdots(1+q^\nu)\cdot(1+q)(1+q^2)\cdots(1+q^\pi)$, with equality for $A=\{-1,\ldots,-\nu\}\cup\{1,\ldots,\pi\}\cup Z$, $Z=\emptyset$ or $\{0\}$. Lemma 5.2 (p. 179) compares the middle coefficients of $G(q)(1+q^{j+1})$ and $G(q)(1+q^j)$. Corollary 5.3 (p. 179): for $n$ distinct reals and subsets with at most $k$ distinct element sums, with $\nu=[(n-1)/2]$ and $\pi=[n/2]$, $r$ is at most the sum of the $k$ middle coefficients of $2(1+q)(1+q^2)\cdots(1+q^\nu)\cdot(1+q)(1+q^2)\cdots(1+q^\pi)$, achieved by $A=\{-\nu,-\nu+1,\ldots,\pi\}$. The paper adds (p. 179) that "The actual conjecture [13, (12)] of Erdös and Moser is equivalent to the case $k=1$, and $n$ odd, of Corollary 5.3", where [13] is Erdős's 1965 survey, and that [35] (G. W. Peck, Erdős' conjecture on sums of distinct numbers, Studies in Applied Math., "to appear") derives the Erdős--Moser conjecture by a purely combinatorial argument from property S of $M(n)$, the $k$-Sperner property for every $k$; p. 178 thanks Ranee Gupta "for pointing out an error in my original treatment of the Erdös--Moser conjecture". The reference list (p. 184) gives [38] as Sárközi and Szemerédi, Acta Arith. 11 (1966), pp. 205--208 (the volume is dated 1965 on its own pages) and [42] as J. H. van Lint, Representation of $0$ as $\sum_{k=-N}^N\varepsilon_kk$, Proc. Amer. Math. Soc. 19 (1967), 182--184. The paper does not cite Halász. That PDF prints "© 1980 Society for Industrial and Applied Mathematics" on its first page; the term recorded, reserved, is read from that copyright notice.

Read status: claims checked for the abstract's statement (p. 168), Corollary 5.1, Lemma 5.2 and Corollary 5.3 (pp. 178--179) and the two remarks on the Erdős--Moser conjecture (pp. 178--179), read clause by clause on the page images and in the text layer on 2026-09-18; the proofs, which rest on Theorem 3.1 (property S of the Bruhat-order posets $W^J$, via the hard Lefschetz theorem) and Proposition 2.5 (on products of varieties with cellular decompositions), were not checked. Nothing here is independently reviewed.

Bears on. #362 (Corollary 5.1, p. 178: with $\nu=\zeta=0$, $k=1$, the exact maximum number of subsets of $N$ distinct positive reals with a common sum, the middle coefficient of $(1+q)(1+q^2)\cdots(1+q^N)$, attained by $\{1,\ldots,N\}$; Corollary 5.3, p. 179: the maximum over all sets of $N$ distinct reals, attained by $\{-[(N-1)/2],\ldots,[N/2]\}$, the set the site's commentary names)

Results to transcribe.

• Lemma 1.1 (p. 168): For a finite graded rank-symmetric poset P of rank n, the following are equivalent: P is rank-unimodal and has property S (the k-Sperner property for every k); P has property T; and there exist order-raising linear maps phi_i: V_i -> V_{i+1} whose composites phi_{n-i-1}...phi_i: V_i -> V_{n-i} are invertible for 0 <= i <= n/2.
• Theorem 2.1 (p. 169): If a complex projective variety X of dimension n has a cellular decomposition {C_i}, the cohomology classes [C_i-bar] form a basis of H^*(X,C), and H^{2m+1}(X,C) = 0 for all m.
• Theorem 2.4 (p. 170): The poset Q^X derived from a cellular decomposition of a nonsingular irreducible complex projective variety X of complex dimension n is graded of rank n, rank-symmetric, rank-unimodal and has the k-Sperner property for every k. Paged as theorem_2_4.
• Theorem 3.1 (p. 172): For a Coxeter system (W, S) with W a Weyl group and J a subset of S, the Bruhat-order poset W^J, which is Q^X for X = G/P, is rank-symmetric, rank-unimodal and has property S. Paged as theorem_3_1.
• Erdos-Moser conjecture: For a set S of 2l+1 distinct reals, the number of subsets of S with equal element sums is at most the middle coefficient of 2(1+q)^2(1+q^2)^2...(1+q^l)^2, and this bound is best possible; deduced from the case G = SO(2n+1) with P a certain maximal parabolic subgroup. This is the abstract's statement (p. 168), the case n = 2l+1 odd, k = 1 of Corollary 5.3.
• Corollary 5.1 (p. 178): For a set A of distinct reals with nu negative elements, zeta zeros and pi positive elements, at most the sum of the k middle coefficients of 2^zeta prod_{i<=nu}(1+q^i) prod_{i<=pi}(1+q^i) subsets of A have element sums taking at most k values, with equality for {-1,...,-nu} union {1,...,pi} (union {0} if zeta = 1). Paged as corollary_5_1.
• Corollary 5.3 (p. 179): For n distinct reals, with nu = [(n-1)/2] and pi = [n/2], at most the sum of the k middle coefficients of 2 prod_{i<=nu}(1+q^i) prod_{i<=pi}(1+q^i) subsets have element sums taking at most k values, with equality for {-nu, -nu+1, ..., pi}. Paged as corollary_5_3.

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