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The dominant observables are in correspondence with some trees, namely rooted trees with vertices of degree at most $D$ and lines colored by a number $i$ from 1 to $D$ such that no two lines connecting a vertex to its descendants have the same color. In this Letter we study by independent methods a generating function for these observables. We prove that the number of such trees with exactly $p_i$ lines of color $i$ is $\\frac{1}{\\sum_{i=1}^D p_i +1} \\binom{\\sum_{i=1}^D p_i+1}{p_1} ... \\binom{\\sum_{i=1"},"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":"1206.4203","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-06-19T13:29:04Z","cross_cats_sorted":["hep-th","math.CO","math.MP"],"title_canon_sha256":"bedf100c95c219f8cf7eb7e4d02aa36e1821b419165298f4e22e016f16770a7e","abstract_canon_sha256":"ea10d7822b8c6f815f3bfcba1eb0ea9d19bc25eff300bb74e53dfbd90cbd7bad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:16.458591Z","signature_b64":"vmWEMHMNxKIsB+69GH51O+DFrpRZT1HKEVk5f6vQ2TJHNIJC7kMdBcdrGHQ9bAqs/U5Nzx/yuWpI6L3I2ZLJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0131c2908a0586293c85eba4002885de450dce487b1402e7c6e6521de1d1c417","last_reissued_at":"2026-05-18T03:53:16.457879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:16.457879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting Line-Colored D-ary Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.CO","math.MP"],"primary_cat":"math-ph","authors_text":"Razvan Gurau, Valentin Bonzom","submitted_at":"2012-06-19T13:29:04Z","abstract_excerpt":"Random tensor models are generalizations of matrix models which also support a 1/N expansion. 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