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The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation

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Pith's one-line read This paper derives exact limiting spectral laws for anticommutators of structured random matrix ensembles, including split-spectrum blip moments for checkerboard pairs.

desk verdict A genuinely useful combinatorial paper whose headline checkerboard blip theorems are conditional on an explicit but unproved spectral-location assumption, plus an unexplained numerical mismatch in Appendix E. read the letter →

arxiv 2502.00505 v2 pith:3I2ZDLMY submitted 2025-02-01 math.PR

Glenn Bruda , Bruce Fang , Raul Marquez , Steven J. Miller , Beni Prapashtica , Vismay Sharan , Daeyoung Son , Saad Waheed
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Janine Wang
This is my paper · ORCID
classification math.PR MSC 60B2015B52
keywords randommatrixtheoryanticommutatorlimitingspectraldistributionGaussianorthogonalensemblepalindromicToeplitzcheckerboardblockcirculantblipregime
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies the anticommutator {A,B}=AB+BA of two independent real symmetric random matrices and asks how the eigenvalue distribution changes when the inputs carry extra symmetry. For Gaussian orthogonal ensemble pairs it obtains exact even moments, M_{2m} = (1/m)\sum_{k=1}^m 2^k \binom{2m}{k-1}\binom{m}{k}, plus an explicit algebraic limiting density; for palindromic Toeplitz pairs it gets M_{2m}=4^m((2m-1)!!)^2, whose density is the convolution of a chi-squared and a negated chi-squared variable. When a checkerboard ensemble is involved, the spectrum splits into regimes of different scales, and the paper isolates the largest outlier regime with a polynomial weight function to get closed-form moment formulas. The payoff is a rare set of explicit benchmarks for structured random matrices, where the usual free-probability anticommutator recipe becomes intractable.

What carries the argument

The load-bearing machinery is the moment method coupled to matching combinatorics. Wick's formula turns expected traces into sums over pairings of matrix entries; for GOE only non-crossing pairings survive in the limit via the genus bound $\#(\gamma_{2m}\pi)\le m-1$ unless $\pi$ is non-crossing, for PTE essentially all pairings survive because of its palindromic structure, and for mixed GOE/PTE a layer decomposition restricts the PTE terms to stay inside layers created by non-crossing GOE matchings. The blip results use the polynomial weight function $f^{(2n)}(x)=x^{2n}(2-x)^{2n}$ with $n=\log\log N$, which is close to 1 at a blip location and decays rapidly elsewhere; expanding the weight and applying binomial identities such as $\sum_i (-1)^i\binom{m}{i}i^p=0$ for $p<m$ cancels all lower-order contributions and leaves the closed-form moments.

What would settle it

For moderately large $N$ with $k \mid N$ and $\gcd(k,j)=1$, $jk \mid N$, diagonalize $\{A_N,B_N\}$ for GOE/$k$-checkerboard and $k$-checkerboard/$j$-checkerboard and count eigenvalues in windows $[N^{3/2}/k - CN, N^{3/2}/k + CN]$ and $[2N^2/(jk)-CN^{3/2}, 2N^2/(jk)+CN^{3/2}]$; the central claim fails if the counts differ from $k$ per sign and $1$, respectively, or if the empirically weighted blip moments do not approach the formulas in Theorems 1.18 and 1.19.

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Extended reading notes

Core claim

The central discovery is that the anticommutator operation preserves enough of the input ensembles' combinatorial structure to yield exact limiting spectral information. For {GOE,GOE} the limiting even moments are the 3-Schr\"oder numbers, and the density has the closed algebraic form given in Corollary 1.10; for {PTE,PTE} the moments factor as 4^m((2m-1)!!)^2 and the density is the convolution of the densities of $\chi_1^2$ and $-\chi_1^2$; for {GOE,PTE} a two-variable recurrence $\sigma_{n,s}=\sum_{k=1}^n (\sigma_{k-1,1}\sigma_{n-k,s}+\sigma_{k-1,0}\sigma_{n-k,s+1})$ governs the moments. When a checkerboard ensemble is involved, the spectrum splits: {GOE,$k$-checkerboard} has a bulk of size $\Theta(N)$ plus a blip of $2k$ eigenvalues near $\pm N^{3/2}/k$, and the weighted blip moments converge to $(1/k)(2/k^2)^m \mathbb{E}_k[\operatorname{Tr} C_k^m]$, where $C_k$ is a $k\times k$ hollow GOE. For {$k$-checkerboard,$j$-checkerboard} there is a largest blip near $2N^2/(jk)$ whose moments have the explicit multinomial formula of Theorem 1.19.

Load-bearing premise

The regime classification for checkerboard anticommutators rests on an unproved empirical input from Appendix B: that the mean-matrix part has exactly $k$ eigenvalues at each of $\pm N^{3/2}/k$, and the analogous counts for the $k,j$ case, so if those positions or multiplicities were wrong the weight functions would isolate the wrong eigenvalues.

Editorial extensions

If this is right

  • If the {GOE,GOE} moment formula is correct, the limiting density is a new explicit algebraic benchmark for anticommutators of structured random matrices.
  • The {PTE,PTE} result implies the limiting spectrum of the anticommutator of two palindromic Toeplitz matrices is the same as the difference of two independent chi-squared variables.
  • The {GOE,$k$-checkerboard} blip formula shows that large-$N$ extreme eigenvalues are governed by a fixed $k\times k$ hollow GOE, making small-matrix computations a proxy for extreme spectral statistics.
  • The {$k$-checkerboard,$j$-checkerboard} largest-blip formula provides an exact moment sequence for a single outlier eigenvalue that can be compared with numerical spectra.
  • The genus expansions for the block circulant anticommutators give a route to compute higher moments numerically even where no closed form is known.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Appendix B spectral-location input were proved rather than observed, the weight-function method would upgrade from conditional to unconditional; a promising route is a deterministic equivalent for the mean-matrix anticommutator.
  • The same weighted-moment cancellation should apply to other split-limiting ensembles with several outlier regimes, provided the outlier locations are known to the correct order; the paper's block taxonomy of 1-blocks and 2-blocks may generalize.
  • The appearance of 3-Schr\"oder numbers in {GOE,GOE} hints at a lattice-path or walk interpretation of anticommutator moments that could connect to enumerative combinatorics beyond the paper.
  • The $k$-dependence of the blip moment $(1/k)(2/k^2)^m \mathbb{E}_k[\operatorname{Tr} C_k^m]$ suggests a universality: only the dimension of the checkerboard's mean-space matters, not the detailed distribution of the non-weight entries, given finite higher moments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the limiting spectral distribution of anticommutators {A,B}=AB+BA of real symmetric random matrix ensembles. For {GOE,GOE}, {PTE,PTE}, and {GOE,PTE}, Section 2 gives closed-form moment formulas or recurrences: Theorem 1.9/2.10 gives M_{2m}=(1/m)\sum_{k=1}^m 2^k \binom{2m}{k-1}\binom{m}{k}; Theorem 1.11/2.12 gives M_{2m}=4^m((2m-1)!!)^2; Theorem 1.13/2.23 gives a recurrence for {GOE,PTE}. For {GOE,k-BCE} and {k-BCE,k-BCE}, Theorem 1.15/2.28 gives genus-expansion formulas. Section 3 introduces weighted empirical blip spectral measures for the anticommutators involving checkerboard ensembles and claims limiting blip moments in Theorems 1.18/3.17 and 1.19/3.20. Appendix B aims to prove the multiple-regime structure, Appendix C gives convergence statements, Appendix D gives explicit weight functions, and Appendix E reports numerical lower moments.

Significance. If the main results hold, the closed-form moment formulas and densities for {GOE,GOE} and {PTE,PTE} are useful additions to the random-matrix literature, and the blip moment formulas for checkerboard anticommutators would provide new benchmarks for non-bulk spectral regimes. The combinatorial machinery in Section 2 is coherent and self-contained: the Wick expansion, the non-crossing matching arguments, the recurrence for {GOE,PTE}, and the identification of the {GOE,GOE} generating function with the 3-Schr\"oder numbers are all presented in a verifiable way. The convolution representation of the {PTE,PTE} density is elegant. However, the blip theorems rest on an explicitly empirical spectral-location input in Appendix B, and Appendix E contains an unexplained numerical discrepancy; these issues are load-bearing for the paper's central new claims.

major comments (3)
  1. [Appendix B, Lemmas B.5 and B.7; Theorems 1.18/3.17 and 1.19/3.20]
  2. [Lemma B.7(1); Definition 3.2 and equation (3.4)]
  3. [Appendix E, Table 4]
minor comments (5)
  1. [Section 1.3, the paragraph after Figure 8]
  2. [Section 3.1, displayed line after equation (3.18)]
  3. [Bibliography reference [NR]]
  4. [Appendix D, equation (D.1)]
  5. [Section 2, theorem numbering]

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the moment computations are self-contained combinatorial counts, while the unproved empirical blip locations in Appendix B are a validity gap rather than a circular input.

full rationale

The central moment claims for {GOE, GOE}, {PTE, PTE}, and {GOE, PTE} are derived by counting non-crossing and free matchings via Wick's formula and the eigenvalue trace lemma; no parameter is fitted to the reported moments, and the identifications with OEIS A027307 / 3-Schroeder numbers and with the Nica-Speicher anticommutator density are external known results. The checkerboard blip theorems are the only place where an unproved spectral input enters: before Lemma B.5 the paper states 'Empirically, we observe that A_N has k blip eigenvalues at N^{3/2}/k + O(N) ... By assuming this, we are able to prove the existence of multiple regimes', and the weight functions in Definitions 1.16 and 1.17 are centered at exactly those empirically asserted locations. This makes Theorems 1.18 and 1.19 conditional on a numerically observed but unproved spectral-location assumption; that is a correctness and completeness concern, not a circular reduction, because the resulting moment formulas are not equivalent to the assumed locations and are obtained from independent combinatorial counts of cyclic products and cancellation identities. The self-citations, chiefly to [BCDHMSTPY] and [MMS], support technical tools such as the weight-function method and the free-matching property of PTE; these tools are either re-proven in the present text or are published external results, so they are not solely load-bearing. The Appendix E discrepancy for {2-BCE, 2-BCE} (theoretical fourth moment 5.52 versus empirical 10.19) signals an internal consistency or implementation problem, but it does not show that any derivation reduces to its own inputs. No step in the paper exhibits Eq. X equal to Eq. Y by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical constants are fitted to data; the ensemble normalizations are fixed by definition. The checkerboard blip locations are not derived in the paper but are empirical inputs assumed in Appendix B. No new physical entities are introduced; the weight functions are analytic devices.

assumptions (6)
  • standard math Wick's formula and the GOE pairing expansion (Proposition 2.3).
    Used throughout Section 2 to reduce expected moments to counts of pairings.
  • standard math Non-crossing pairings dominate GOE moments (Proposition 2.4 from [MS]).
    Invoked in Lemma 2.5 and Theorem 2.6 to identify the limiting contributions.
  • domain assumption Free matching property of PTE from [MMS].
    Used in Section 2.2 to count all pairings with consistent indexing; this is a non-elementary prior result.
  • domain assumption k-BCE matching relations from [KKMSX].
    Used in Section 2.4 for the genus expansion formulas for {GOE,k-BCE} and {k-BCE,k-BCE}.
  • ad hoc to paper Empirical observation that anticommutators involving checkerboard mean matrices have blip eigenvalues at given locations, including N^{3/2}/k and related scales.
    Appendix B explicitly says "Empirically, we observe... By assuming this, we are able to prove..." This is load-bearing for the existence of checkerboard blip regimes.
  • domain assumption Moment convergence implies weak convergence of the spectral measures via fourth-moment bounds from [BCDHMSTPY] and [MMS].
    Appendix C relies on this to pass from limiting expected moments to almost sure weak convergence.

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Pith. "Pith review of The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation." pith.science (2026). https://pith.science/paper/3I2ZDLMY

@misc{pith2026250200505,
  author       = {Pith},
  title        = {Pith review of: The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3I2ZDLMY}},
  note         = {Machine review of arXiv:2502.00505}
}
abstract

Inspired by the quantization of classical quantities and Rankin Selberg convolution, we study the anticommutator operation $\{\cdot, \cdot\}$, where $\{A,B\} = AB + BA$, applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the $k$-checkerboard ensemble, and the block $k$-circulant ensemble ($k$-BCE). Using combinatorial and topological techniques related to non-crossing and free matching properties of GOE and PTE, we obtain closed-form formulae for the moments of the limiting spectral distributions of $\{$GOE, GOE$\}$, $\{$PTE, PTE$\}$, $\{$GOE, PTE$\}$ and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, $\{$GOE, $k$-checkerboard$\}$ and $\{$$k$-checkerboard, $j$-checkerboard$\}$ exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of $\{$GOE, $k$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$ and 1 blip regime of size $\Theta(N^{3/2})$, the spectrum of $\{$$k$-checkerboard, $j$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$, 2 intermediary blip regimes of size $\Theta(N^{3/2})$, and 1 largest blip regime of size $\Theta(N^2)$. In both cases, with the appropriate weight function, we are able to isolate the largest regime for other regime(s) and analyze its moments and convergence results via combinatorics. We end with numerical computation of lower even moments of $\{$GOE, $k$-BCE$\}$ and $\{$$k$-BCE, $k$-BCE$\}$ based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of $\{$$k$-checkerboard, $j$-checkerboard$\}$.

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Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Basak and A

    A. Basak and A. Bose, Balanced random Toeplitz and Hankel matrices, Electronic Comm. in Prob. 15 (2010), 134--148

    work page 2010

  2. [2]

    Basak and A

    A. Basak and A. Bose, Limiting spectral distribution of some band matrices, Periodica Mathematica Hungarica 63 (2011), no. 1, 113--150

    work page 2011

  3. [3]

    Blackwell, N

    K. Blackwell, N. Borde, C. Devlin VI, N. Luntzlara, R. Ma, S. J. Miller, M. Wang, and W. Xu, Distribution of Eigenvalues of Random Real Symmetric Block Matrices, (2019), submitted

    work page 2019

  4. [4]

    Burkhardt, P

    P. Burkhardt, P. Cohen, J. DeWitt, M. Hlavacek, S. J. Miller, C. Sprunger, Y. N. T. Vu, R. V. Peski, and K. Yang, Random matrix ensembles with split limiting behavior, Random Matrices: Theory and Applications 7(3) (2018), 1850006

    work page 2018

  5. [5]

    Barrett, F

    O. Barrett, F. W. K. Firk, S. J. Miller, and C. Turnage-Butterbaugh, From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond, in Open Problems in Mathematics (eds.\ J. Nash Jr.\ and M. Th.\ Rassias), Springer-Verlag, 2016, 123--171

    work page 2016

  6. [6]

    Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 3rd edition, 1995, 608 pages

    P. Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 3rd edition, 1995, 608 pages

    work page 1995

  7. [7]

    A. Bose, R. S. Hazra, and K. Saha, Patterned random matrices and notions of independence, Technical report R3/2010 (2010), Stat-Math Unit, Kolkata

    work page 2010

  8. [8]

    A. Bose, R. S. Hazra, and K. Saha, Patterned random matrices and method of moments, in Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, 2203--2230 (Invited article). World Scientific, Singapore and Imperial College Press, UK

    work page 2010

Show all 42 references
  1. [9]

    Beckwith, V

    O. Beckwith, V. Luo, S. J. Miller, K. Shen, and N. Triantafillou, Distribution of eigenvalues of weighted, structured matrix ensembles, Integers: Electronic Journal of Combinatorial Number Theory 15 (2015), paper A21, 28 pages

    2015

  2. [10]

    Burstein and L

    A. Burstein and L. W. Shapiro, Pseudo-Involutions in the Riordan Group, Journal of Integer Sequences 25 (2022), Article 22.3.6

    2022

  3. [11]

    F. P. Cantelli, Sulla probabilità come limite della frequenza, Atti Accad. Naz. Lincei 26(1) (1917), 39--45

    1917

  4. [12]

    F. Chen, Y. Lin, S. J. Miller, and J. Yu, The Limiting Spectral Measure for an Ensemble of Generalized Checkerboard Matrices, The PUMP Journal of Undergraduate Research 4 (2021), 202--221

    2021

  5. [13]

    Couillet and M

    R. Couillet and M. Debbah, Random matrix methods for wireless communications, Cambridge University Press, 2011

    2011

  6. [14]

    T. Dunn, H. Fleischmann, F. Jackson, S. Khunger, S. J. Miller, L. Reifenberg, A. Shashkov, and S. Willis, Limiting Spectral Distributions of Families of Block Matrix Ensembles, The PUMP Journal of Undergraduate Research 5 (2022), 122--147

    2022

  7. [15]

    Ferge, Moment equalities for sums of random variables via integer partitions and Faà di Bruno’s formula, Turkish Journal of Mathematics 38(3) (2014), 558--575

    D. Ferge, Moment equalities for sums of random variables via integer partitions and Faà di Bruno’s formula, Turkish Journal of Mathematics 38(3) (2014), 558--575

    2014

  8. [16]

    F. W. K. Firk and S. J. Miller, Nuclei, Primes and the Random Matrix Connection, Symmetry 1 (2009), 64--105

    2009

  9. [17]

    P. J. Forrester, Log-Gases and Random Matrices, London Mathematical Society Monographs 34, Princeton University Press, Princeton, NJ, 2010

    2010

  10. [18]

    P. J. Forrester, Log-gases and random matrices, Number 34, Princeton University Press, 2010

    2010

  11. [19]

    Goldmakher, C

    L. Goldmakher, C. Khoury, S. J. Miller, and K. Ninsuwan, On the spectral distribution of large weighted random regular graphs, to appear in Random Matrices: Theory andApplications. http://arxiv.org/abs/1306.6714 http://arxiv.org/abs/1306.6714

    arXiv

  12. [20]

    Horn and C

    R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985

    1985

  13. [21]

    Hammond and S

    C. Hammond and S. J. Miller, Distribution of Eigenvalues of Real Symmetric Toeplitz Matrices, Journal of Theoretical Probability 18 (2005)

    2005

  14. [22]

    Harer and D

    J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85(3) (1986), 457--485

    1986

  15. [23]

    G. Kopp, F. Strauch, and W. Xiong, The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices, Journal of Theoretical Probability 26(4) (2013), 1020--1060

    2013

  16. [24]

    McKay, The expected eigenvalue distribution of a large regular graph, Linear Algebra Appl

    B. McKay, The expected eigenvalue distribution of a large regular graph, Linear Algebra Appl. 40 (1981), 203--216

    1981

  17. [25]

    Meckes, The spectra of random abelian G-circulant matrices, ALEA Lat

    M. Meckes, The spectra of random abelian G-circulant matrices, ALEA Lat. Am. J. Probab. Math. Stat. 9(2) (2012), 435--450

    2012

  18. [26]

    M. L. Mehta, Random Matrices, Volume 142, Academic Press, 2004

    2004

  19. [27]

    Mezzadri and N

    F. Mezzadri and N. C. Snaith, Recent Perspectives in Random Matrix Theory and Number Theory, Cambridge University Press, 2005

    2005

  20. [28]

    S. J. Miller and R. Morrison, Modeling Convolutions of L-functions, arXiv preprint arXiv:1011.0229 https://arxiv.org/abs/1011.0229, 2010

    2010 arXiv

  21. [29]

    Massey, S

    A. Massey, S. J. Miller, and J. Sinsheimer, Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices, Journal of Theoretical Probability 20 (2007), 637--662

    2007

  22. [30]

    A. J. Mingo and R. Speicher, Free Probability and Random Matrices, Vol. 35, Springer, New York, 2017

    2017

  23. [31]

    S. J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, Princeton, NJ, 2006, 503 pages

    2006

  24. [32]

    Nica and S

    A. Nica and S. Roland, Commutators of free random variables, (1998), 553--592

    1998

  25. [33]

    Rudnick and P

    Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996), 269--322

    1996

  26. [34]

    J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd edition, Cambridge University Press, 2017

    2017

  27. [35]

    Takacs, A Moment Convergence Theorem, The Amer

    L. Takacs, A Moment Convergence Theorem, The Amer. Math. Monthly 98(8) (1991), 742--746

    1991

  28. [36]

    Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics, Vol

    T. Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics, Vol. 126, American Mathematical Society, Providence, p. 195, 2011

    2011

  29. [37]

    T. Tao, 254a, notes 4: The semi-circular law, https://terrytao.wordpress.com/2010/02/02/254a-notes-4-the-semi-circular-law/ https://terrytao.wordpress.com/2010/02/02/254a-notes-4-the-semi-circular-law/, Posted: 2010-02-02, Accessed: 2016-08-04

    2010

  30. [38]

    Vasilchuk, On the asymptotic distribution of the commutator and anticommutator of random matrices, Journal of Mathematical Physics 44(4) (2003), 1882--1908

    V. Vasilchuk, On the asymptotic distribution of the commutator and anticommutator of random matrices, Journal of Mathematical Physics 44(4) (2003), 1882--1908

    2003

  31. [39]

    Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc

    E. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Phil. Soc. 47 (1951), 790--798

    1951

  32. [40]

    Wigner, Statistical Properties of Real Symmetric Matrices, in Canadian Mathematical Congress Proceedings, University of Toronto Press, Toronto, 1957, 174--184

    E. Wigner, Statistical Properties of Real Symmetric Matrices, in Canadian Mathematical Congress Proceedings, University of Toronto Press, Toronto, 1957, 174--184

    1957

  33. [41]

    Wishart, The generalized product moment distribution in samples from a normal multivariate population, Biometrika 20A (1928), 32--52

    J. Wishart, The generalized product moment distribution in samples from a normal multivariate population, Biometrika 20A (1928), 32--52

    1928

  34. [42]

    oder paths and m -Schr\

    S. Yang and M. Jiang, The m -Schr\"oder paths and m -Schr\"oder numbers, Discrete Mathematics 344(2) (2021), 112209

    2021

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