Explicit tableau-pair families for the Rogers–Ramanujan summands and the direct Rogers–Ramanujan bijection (Stanton's Problem 2.3, with the row-restricted Cauchy Speculation 4.2)

Stanton's 2025 note "The Rogers–Ramanujan Identities and Cauchy's Identity" (SIGMA 21 (2025), 049) reduces the century-old search for a direct bijective proof of the Rogers–Ramanujan identities to two concrete open questions about Schur functions and the Robinson–Schensted–Knuth (RSK) correspondence. Problem 2.3 asks for an explicit, uniform choice, for every $n$, of a family of pairs of column-strict tableaux $(P,Q)$ of the same shape (entries of $P$ multiples of 5, entries of $Q$ equal to 1 or 4) whose generating function is the $n$-th Rogers–Ramanujan summand $F_n(q)=q^{n^2}/(q;q)_n$; composed with the paper's RSK-based bijection this would yield the long-sought natural Rogers–Ramanujan bijection between partitions into parts congruent to 1 or 4 modulo 5 and partitions with parts differing by at least 2. Speculation 4.2 asks whether the two-set, row-restricted Cauchy sum at modulus $2k+3$ admits a finite decomposition into products of infinite $q$-products. Both questions are currently open.

Background

A partition of an integer $m$ is a nonincreasing sequence of positive integers summing to $m$. The first Rogers–Ramanujan identity states

$$\sum_{n\ge0}\frac{q^{n^{2}}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty},$$

where $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^{j})$ is the $q$-Pochhammer symbol (and $(a;q)_\infty$ its infinite limit). MacMahon's combinatorial reading equates two partition families: the left side's $n$-th summand $q^{n^{2}}/(q;q)_n$ generates partitions with exactly $n$ parts that differ by at least 2 ("super-distinct" or Rogers–Ramanujan partitions), while the right side generates partitions whose parts are congruent to 1 or 4 modulo 5. After the analytic proofs of Rogers, Ramanujan, and Schur, a direct bijective proof — an explicit size-preserving correspondence between the two families whose bijectivity is evident without cancellation — became and remains a famous open demand. Garsia and Milne (1981) constructed a bijection using their involution principle, a chain of sign-reversing involutions, but it is universally regarded as indirect; Pak's survey and Sills's book both record the natural Rogers–Ramanujan bijection as a leading open problem, and Andrews's classic monograph collects the analytic theory.

Stanton's 2025 note recasts the question through symmetric functions. A semistandard (column-strict) tableau of partition shape $\lambda$ fills the Young diagram of $\lambda$ with integers that weakly increase along rows and strictly increase down columns; the Schur function $s_\lambda(x_1,x_2,\ldots)$ is the weight-generating function of these fillings (Stanton's constructions also allow the entry 0, as in his Tables 1–2). The Cauchy identity $\sum_\lambda s_\lambda(x)s_\lambda(y)=\prod_{i,j}(1-x_iy_j)^{-1}$ sums over all shapes, and Stanton studies what happens when the number of rows is restricted on each side. Through the Robinson–Schensted–Knuth (RSK) correspondence — the classical bijection between (weighted) two-line arrays and pairs of semistandard tableaux of the same shape — Stanton's Proposition 2.2 maps partitions into parts $\equiv1,4\pmod5$ to pairs $(P,Q)$ of same-shape semistandard tableaux in which the entries of $P$ are multiples of 5 (0 included) and the entries of $Q$ are 1 or 4. What is missing for a direct Rogers–Ramanujan bijection is a choice, uniform in $n$, of which such pairs to take so that the family indexed by $n$ has generating function equal to $F_n(q)=q^{n^{2}}\left(\prod_{j=1}^{n}\sum_{p=0}^{4}q^{jp}\right)/(q^5;q^5)_n$, which simplifies to $q^{n^{2}}/(q;q)_n$; this is Problem 2.3, and the paper exhibits solutions only for small $n$ (its Tables 1–2), remarking that resolving it for arbitrary $n$ "will take substantially more insight". In the same framework, Stanton's Theorem 4.1 proves the $R=1$ case of a general structural speculation (Speculation 4.2) about finite decompositions of row-restricted two-set Cauchy sums into infinite $q$-products at modulus $2k+3$, the case $R=2k$ being the unrestricted Cauchy identity, while the intermediate range is left open. Related but distinct machinery exists: Stembridge's Hall–Littlewood treatment of Rogers–Ramanujan-type identities, and Rains and Warnaar's bounded Littlewood identities, which concern single-set sums with bounded rows or columns and generally pay a determinant-shaped price rather than addressing Stanton's two-set row-restricted sums.

Problem Statement

The problem has one central objective and one companion structural question, both posed as open in Stanton (SIGMA 21 (2025), 049). A complete solution must resolve both: the central objective (a) and the companion structural question (b).

(a) Central objective — Problem 2.3 and the direct Rogers–Ramanujan bijection. Construct, explicitly and uniformly in $n$, a family $\mathcal{S}_n$ of pairs $(P,Q)$ of column-strict (semistandard) tableaux of the same shape, such that the entries of $P$ are multiples of 5 (the entry 0 is allowed, as in the source's Tables 1–2) and the entries of $Q$ are 1 or 4, and the generating function of $\mathcal{S}_n$ by total sum of entries equals

$$F_n(q)=q^{n^{2}}\left(\prod_{j=1}^{n}\sum_{p=0}^{4}q^{jp}\right)\Big/\left(q^{5};q^{5}\right)_{n}=\frac{q^{n^{2}}}{(q;q)_{n}}.$$

The construction must be a single explicit rule (or algorithm) valid for all $n\ge1$ — not a mere existence argument, which follows by counting from the Rogers–Ramanujan identity itself — and, composed with the RSK-based bijection of Proposition 2.2 of the source, must yield a direct natural bijection between, on one side, partitions into parts congruent to 1 or 4 modulo 5 and, on the other, partitions whose parts differ by at least 2, preserving total size.

(b) Companion question — Speculation 4.2. Let $k\ge1$ and $1\le i\le2k+2$, and specialize the variables as $(x_1,x_2,\ldots)=(1,q^{2k+3},q^{2(2k+3)},\ldots)$ and $(y_1,\ldots,y_{2k})=(q^1,\ldots,q^{2k+2})$ with the two entries $q^{i}$ and $q^{2k+3-i}$ deleted. For a row bound $R$ with $1\le R\le2k$, decide whether the row-restricted two-set Cauchy sum

$$\sum_{\lambda\text{ with at most }R\text{ rows}}s_{\lambda}(x_1,x_2,\ldots)\,s_{\lambda}(y_1,\ldots,y_{2k})$$

is a finite sum of terms, each equal to a product of $R$ infinite products of the form $1/(q^{j};q^{2k+3})_{\infty}$ with $j\not\equiv\pm i,0\pmod{2k+3}$, multiplied by coefficients that are rational functions of $q$. The case $R=1$ is proved (Theorem 4.1 of the source) and the case $R=2k$ is the unrestricted Cauchy identity; the intermediate range $2\le R\le2k-1$ is open.

References

1. Dennis Stanton, "The Rogers–Ramanujan Identities and Cauchy's Identity", Symmetry, Integrability and Geometry: Methods and Applications 21 (2025), 049, 6 pages, DOI 10.3842/SIGMA.2025.049, arXiv:2507.00433, https://arxiv.org/abs/2507.00433 (source paper)
2. A. M. Garsia and S. C. Milne, "A Rogers-Ramanujan bijection", Journal of Combinatorial Theory, Series A 31 (1981) 289–339, DOI 10.1016/0097-3165(81)90062-5, https://doi.org/10.1016/0097-3165(81)90062-5
3. Igor Pak, "Partition bijections, a survey", The Ramanujan Journal 12 (2006) 5–75, DOI 10.1007/s11139-006-9576-1, https://doi.org/10.1007/s11139-006-9576-1
4. Andrew V. Sills, "An Invitation to the Rogers-Ramanujan Identities", Chapman and Hall/CRC (2017), DOI 10.1201/9781315151922, https://doi.org/10.1201/9781315151922
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6. John R. Stembridge, "Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities", Transactions of the American Mathematical Society 319 (1990) 469–498, DOI 10.1090/S0002-9947-1990-0986702-5, https://doi.org/10.1090/S0002-9947-1990-0986702-5
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