Pith Number

pith:MWBFHCDL

pith:2026:MWBFHCDL2U7JWI33HN5YRSZEXX

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The Cullis' determinant as Pfaffian

Alexander Guterman, Andrey Yurkov

The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition.

arxiv:2605.14010 v1 · 2026-05-13 · math.CO

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Claims

C1strongest claim

we express the Cullis' determinant of a matrix X as the Pfaffian of the matrix obtained from X by matrix multiplication and transposition

C2weakest assumption

The specific matrix construction via multiplication and transposition produces a skew-symmetric matrix whose Pfaffian exactly matches the alternating sum of maximal minors for arbitrary rectangular X.

C3one line summary

The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.

References

20 extracted · 20 resolved · 1 Pith anchors

[1] A. Amiri, M. Fathy, and M. Bayat. Generalization of some determinantal identities for non-square matrices based on Radic’s definition.TWMS J. Pure Appl. Math., 1(2):163– 175, 2010 2010

[2] Dover Books on Mathematics 2016

[3] C. E. Cullis.Matrices and Determinoids: Volume 1. Calcutta University Readership Lectures. Cambridge University Press, 1913 1913

[4] W. Fulton and P. Pragacz.Schubert Varieties and Degeneracy Loci. Springer Berlin Heidelberg, 1998 1998

[5] G. Galbiati and F. Maffioli. On the computation of pfaffians.Discrete Applied Mathe- matics, 51(3):269–275, 1994 1994

Receipt and verification

First computed	2026-05-17T23:39:13.072715Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

658253886bd53e9b237b3b7b88cb24bdec510c0c45fa515d617e5b7b2f605074

Aliases

arxiv: 2605.14010 · arxiv_version: 2605.14010v1 · doi: 10.48550/arxiv.2605.14010 · pith_short_12: MWBFHCDL2U7J · pith_short_16: MWBFHCDL2U7JWI33 · pith_short_8: MWBFHCDL

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MWBFHCDL2U7JWI33HN5YRSZEXX \
  | 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: 658253886bd53e9b237b3b7b88cb24bdec510c0c45fa515d617e5b7b2f605074

Canonical record JSON

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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T18:21:43Z",
    "title_canon_sha256": "6b03feb09e1589eae8e065bf1ac6675d810c56831afad45d91d6739302e16168"
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