OpenFrontiers
Join
← Back to the index
OP-9584Status unassessedCombinatorics

Tableaux problem 24

Tableaux problem 24

MSC 05 · Combinatorics

Inferred from the community field; finer classification has not been reviewed.

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

24.[Stan1999a]*Ch. 6 Exer. 25(i) Let the symmetric group Sn\mathfrak{S}_{n} act on the polynomial ring A=C[x1,…,xn,y1,…,yn]A=\mathbb{C}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}] by permuting both the x∙x_{\bullet} and the y∙y_{\bullet} simultaneously; w⋅f(x1,…,xn,y1,…,yn)=f(xw(1),…,xw(n),yw(1),…,yw(n))w\cdot f(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})=f(x_{w(1)},\ldots,x_{w(n)},y_{w(1)},\ldots,y_{w(n)}) for all w∈Snw\in\mathfrak{S}_{n}. Let II be the ideal generated by all invariants of positive degree, i.e.,

I=<f∈A:w⋅f=f for all w∈Sn, and f(0)=0>.I=\left<f\in A:w\cdot f=f\textup{ for all }w\in\mathfrak{S}_{n}\textup{, and }f(0)=0\right>.

Prove that the Catalan number Cn=1n+1(2nn)C_{n}=\frac{1}{n+1}\binom{2n}{n} is the dimension of the subspace of A/IA/I affording the sign representation, i.e.,

Cn=dim⁡{f∈A/I:w⋅f=(sgn⁡w)f for all w∈Sn}.C_{n}=\operatorname{dim}\{f\in A/I:w\cdot f=(\operatorname{sgn}w)f\textup{ for all }w\in\mathfrak{S}_{n}\}.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Dale R. Worley, version 3 (5 April 2026). Source citation keys are retained even where the arXiv HTML does not resolve its bibliography.

CC BY 4.0; Dale R. Worley. HTML converted to Markdown with attribution and source links.

References
  1. [1]Tableaux — A notebook of open problems — Tableaux problem 24 link
  2. [2]Creative Commons Attribution 4.0 International link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.