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Let s(n,m) denote a permutation chosen uniformly at random among all permutations of [n] with exactly m inversions. Let p(n,m) be the common value for the probabilities that s(n,m) is indecomposable or the corresponding graph is connected. We prove that p(n,m) is non-decreasing with m by constructing a Markov process in which s(n,m+1) is obtained from s(n,m) by increasing one of the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.0246","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-01T18:11:25Z","cross_cats_sorted":[],"title_canon_sha256":"fa7dccd09ec7d23931ab86c989d29d5e18f54ee6c3a8565b674f10f339ebb5a5","abstract_canon_sha256":"affbf74389aa536728fbb875883cd62942393d6e5f70ebf03bfd30393ff38773"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:00.246959Z","signature_b64":"q76u9bsCp7C7UWsaivk5qG03dRypJyZy7dU3aI3YhDNs/8FOarr7rrGCAGga2zbpx/GBGV+r4+1H/dpNurXnCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e24e653f6fd9454c9da69306b4ef15e3499333bc09ff83a0d4da1342cf905b1a","last_reissued_at":"2026-05-18T03:13:00.246047Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:00.246047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the connected components of a random permutation graph with a given number of edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boris Pittel, Huseyin Acan","submitted_at":"2012-11-01T18:11:25Z","abstract_excerpt":"A permutation of [n] induces a graph on [n] such that the edges of the graph correspond to inversion pairs of the permutation. 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