Pith Number

pith:MBJQSZ53

pith:2026:MBJQSZ53T4FSBOYW3LRUQNPKRI

not attested not anchored not stored refs resolved

Boolean--Eulerian numbers

Mikl\'os B\'ona, Vincent Vatter

The number of red- or blue-colored decreasing binary trees equals 2^{n-1} times the nth Euler number.

arxiv:2605.15415 v1 · 2026-05-14 · math.CO

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\pithnumber{MBJQSZ53T4FSBOYW3LRUQNPKRI}

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4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

The count equals 2^{n-1} times the nth Euler number; the Boolean-Eulerian polynomials are an explicit algebraic transform of the classical Eulerian polynomials; the Foata-Strehl orbit decomposition recast in the decreasing-binary-tree model gives a direct combinatorial proof of gamma-positivity.

C2weakest assumption

The two constructed bijections are valid and preserve all the required statistics (odd block sizes, alternating permutations, right edges, binary labels). This premise is invoked in the statements of the two bijections and in the refinement by right edges (abstract, paragraphs describing the first and second bijections).

C3one line summary

Boolean-Eulerian numbers count red-blue colored decreasing binary trees and equal 2^{n-1} times the nth Euler number; their polynomials are algebraic transforms of Eulerian polynomials that inherit gamma-positivity, real-rootedness, and zero interlacing.

References

11 extracted · 11 resolved · 1 Pith anchors

[1] Boolean-Narayana numbers · arXiv:2602.11355

[2] Stack-sorting preimages and 0 - 1 -trees

[3] Unimodal, log-concave and P \'olya frequency sequences in combinatorics 1989

[4] The applications of total positivity to combinatorics, and conversely 1994

[5] Th\'eorie g\'eom\'etrique des polyn\^omes eul\'eriens , vol 1970

Receipt and verification

First computed	2026-05-20T00:00:57.406634Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

60530967bb9f0b20bb16dae34835ea8a18d2bd750d2dc05cac492a0db0dad195

Aliases

arxiv: 2605.15415 · arxiv_version: 2605.15415v1 · doi: 10.48550/arxiv.2605.15415 · pith_short_12: MBJQSZ53T4FS · pith_short_16: MBJQSZ53T4FSBOYW · pith_short_8: MBJQSZ53

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MBJQSZ53T4FSBOYW3LRUQNPKRI \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 60530967bb9f0b20bb16dae34835ea8a18d2bd750d2dc05cac492a0db0dad195

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "317adc392ec63e577d63717a34596a9b063d0012549e91530a518c4d4ed87081",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-14T20:59:45Z",
    "title_canon_sha256": "29abb792a04b250856929d2c6d0cf2a26764d89c9566de189dd5b77534084b2d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15415",
    "kind": "arxiv",
    "version": 1
  }
}