{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NFL6EWVD2665SJM3VIZN7DB6AJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0815bd906d22982443999afdfab7e4615e09f16e666e490fa771c00d979b3dce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-01-27T16:37:30Z","title_canon_sha256":"2e3205595233159fc5e66e2c2f460e7b82dd2217e8df451e1b193a1ff50b0ff9"},"schema_version":"1.0","source":{"id":"1201.5825","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.5825","created_at":"2026-05-18T04:01:25Z"},{"alias_kind":"arxiv_version","alias_value":"1201.5825v2","created_at":"2026-05-18T04:01:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.5825","created_at":"2026-05-18T04:01:25Z"},{"alias_kind":"pith_short_12","alias_value":"NFL6EWVD2665","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NFL6EWVD2665SJM3","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NFL6EWVD","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:06d1183928cc0987db2761ee6056086ead0ecc7f54016b76395e5c722f24b93b","target":"graph","created_at":"2026-05-18T04:01:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We derive a formula for the moments and the free cumulants of the multiplication of $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution $\\mu^{\\boxtimes k}$, given by Kargin which show that the support grows at most linearly with $k$. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theo","authors_text":"Carlos Vargas, Octavio Arizmendi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-01-27T16:37:30Z","title":"Products of free random variables and k-divisible partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5825","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fbe62b1c788de27d9e021e6a8424e3c00be9608626cf6a2e2d714408bee18540","target":"record","created_at":"2026-05-18T04:01:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0815bd906d22982443999afdfab7e4615e09f16e666e490fa771c00d979b3dce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-01-27T16:37:30Z","title_canon_sha256":"2e3205595233159fc5e66e2c2f460e7b82dd2217e8df451e1b193a1ff50b0ff9"},"schema_version":"1.0","source":{"id":"1201.5825","kind":"arxiv","version":2}},"canonical_sha256":"6957e25aa3d7bdd9259baa32df8c3e0268ffb577d8ec3ebca616ee32108f71ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6957e25aa3d7bdd9259baa32df8c3e0268ffb577d8ec3ebca616ee32108f71ed","first_computed_at":"2026-05-18T04:01:25.625442Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:25.625442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sQXoPSCjbFWzcFGy44/lxZPqa1AZmV538vLd1+InrsVuRL+MzI2pd6z4V2BgcOtAxK1Oxk8xr60rJlUyIALHBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:25.626145Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.5825","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fbe62b1c788de27d9e021e6a8424e3c00be9608626cf6a2e2d714408bee18540","sha256:06d1183928cc0987db2761ee6056086ead0ecc7f54016b76395e5c722f24b93b"],"state_sha256":"5edfb9f53d4b5b911a50ee420f824fdb470b4270e93dc6e644b3e2da8b32a3e6"}