benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let n be a nonnegative integer. Determine the number of ways that one can choose (n+1)² sets S_(i,j)⊆{1,2,…,2n} , for integers i,j with 0≤i,j≤n , such that: 1. for all 0≤i,j≤n , th…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let nn be a nonnegative integer. Determine the number of ways that one can choose (n+1)2(n+1)^2 sets Si,j⊆{1,2,…,2n}S_{i,j}\subseteq\{1,2,\ldots,2n\} , for integers i,ji,j with 0≤i,j≤n0\leq i,j\leq n , such that: 1. for all 0≤i,j≤n0\leq i,j\leq n , the set Si,jS_{i,j} has i+ji+j elements; and 2. Si,j⊆Sk,lS_{i,j}\subseteq S_{k,l} whenever 0≤i≤k≤n0\leq i\leq k\leq n and 0≤j≤l≤n0\leq j\leq l\leq n . Contents 1 Solution 1 2 Solution 2 2.1 Lemma 2.2 Filling in the rest of the grid 2.3 Finishing off 3 See also
Plain-text mathematical notation (without MathML)
Let n be a nonnegative integer. Determine the number of ways that one can choose (n+1)² sets S_(i,j)⊆{1,2,…,2n} , for integers i,j with 0≤i,j≤n , such that:
 1. for all 0≤i,j≤n , the set S_(i,j) has i+j elements; and
2. S_(i,j)⊆S_(k,l) whenever 0≤i≤k≤n and 0≤j≤l≤n .
 Contents   1  Solution 1  2  Solution 2   2.1  Lemma  2.2  Filling in the rest of the grid  2.3  Finishing off    3  See also
Original LaTeX notation
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$ , for integers $i,j$ with $0\leq i,j\leq n$ , such that:
 1. for all $0\leq i,j\leq n$ , the set $S_{i,j}$ has $i+j$ elements; and
2. $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$ .
 Contents   1  Solution 1  2  Solution 2   2.1  Lemma  2.2  Filling in the rest of the grid  2.3  Finishing off    3  See also

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
49bcfe36-5ec1-52d8-b7bc-066b16479824
task_key
test--49bcfe36-5ec1-52d8-b7bc-066b16479824
task_revision_id
1
config / split
/ test

Official source

initial import