January 2025
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Let be positive integers and be the set of integers modulo n. A conjecture of Baranyai from 1974 asks for a decomposition of k-element subsets of into particular families of sets called "wreaths". We approach this conjecture from a new algebraic angle by introducing the key object of this paper, the wreath matrix M. As our first result, we establish that Baranyai's conjecture is equivalent to the existence of a particular vector in the kernel of M. We then employ results from representation theory to study M and its spectrum in detail. In particular, we find all eigenvalues of M and their multiplicities, and identify several families of vectors which lie in the kernel of M.
![Figure 4: An illustration of the left-hand side of equation (1). Next, we claim that the pairs (W, i) are in bijection with NE upper paths from (l−k−1, d+m−l−k) to (0, d) which visit the diagonal y = x in at least one point. The bijection is given as follows. Given a pair (W, i), consider the sequence of k + l − m North and k − l East steps corresponding to W . After the step at which W reaches (−i, −i), add an additional East step. Denote by W ′ the path corresponding to the longer sequence of North and East steps starting at (l −k −1, d+m−l −k). Observe that W ′ consists of k + l − m North steps and k − l + 1 East steps, never visits a point below the diagonal y = x but visits a point on this diagonal. Moreover, the first visited point on this diagonal is (−i, −i). On the other hand, let W ′ be a path from (l − k − 1, d + m − l − k) to (0, d) consisting of North and East steps that never goes below the diagonal y = x but which visits this diagonal in at least one point. Then there is a first point at which W ′ visits this diagonal; let this point be (−i, −i). Consider the sequence of k + l − m North and k − l + 1 East steps corresponding to W ′ . Remove the East step that leads to (−i, −i) to obtain a shorter sequence of steps, and denote by W the corresponding path starting at (l − k, d + m − l − k). Then (W, i) is a pair where i ∈ [k + l − m − d] and W is a NE upper path from (l − k, d + m − l − k) to (0, d) which goes through the point (−i, −i). The maps from the previous two paragraphs are inverses to each other, and hence describe the desired bijection, see Figure 5 for an illustration. We thus obtain that the left-hand side of equation (1) counts the number of NE upper paths from (l − k − 1, d + m − l − k) to (0, d) which visit the diagonal y = x in at least one point. Note that such paths are exactly the NE upper paths from (l − k − 1, d + m − l − k) to (0, d) which visit a point below the diagonal y = x + 1. The number of such paths is P](https://www.researchgate.net/publication/387975860/figure/fig4/AS:11431281303325639@1736826177902/An-illustration-of-the-left-hand-side-of-equation-1-Next-we-claim-that-the-pairs-W_Q320.jpg)
