{"paper":{"title":"Eigenvalues and Eigenvectors of the Matrix of Permutation Counts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Pawan Auorora, Shashank K Mehta","submitted_at":"2013-09-16T06:40:15Z","abstract_excerpt":"Define a $(n^4+n^2)/2\\times (n^4+n^2)/2$ symmetric $B$. $(ij)(kl)$ is an index where $i,j,k,l\\in [n]$, $(ab)$ is an unordered pair and $(kl)$ is an ordered pair when $i\\neq j$, otherwise it is also an unordered pair. $B((ij)(kl),(ab)(xy))$ is equal to the number of permutations of S_n in which $\\min\\{i,j\\}$ maps to $k$, $\\max\\{i,j\\}$ maps to $l$, $\\min\\{a,b\\}$ maps to $x$ and $\\max\\{a,b\\}$ maps to $y$. We will show that $B$ has four distinct eigenvalues: $(3/2)n!$, $n(n-3)!$, $(n-1)!/(n-3)$, $2n(n-2)!$ and the corresponding eigenspace dimensions are 1, ${{n-1}\\choose{2}}^2$, $({{n-1}\\choose{2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3836","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}