{"paper":{"title":"Brick Wall Excursions: Combinatorial Interpretation of Random Flight Moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Khaydar Nurligareev, Michael Wallner, Sergey Kirgizov","submitted_at":"2026-05-15T23:58:29Z","abstract_excerpt":"We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$, the $n$th moment can be interpreted as the number of abelian squares of length $2n$ over an alphabet with $m$ letters, for $d=4$ no interpretation was known.\n  The goal of this paper is to provide such an interpretation, both for $d=2$ and $d=4$, in terms of $2n$-step lattice paths in dimension $m-1$. Our construction relies on a bijection between Dyck paths w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16715","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16715/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.355326Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.479188Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3906aab39f9224d842e4593b30dee3016ae64c1c0ef6201f572a93bcf61ee07d"},"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"}