Symplectic lattice counting and zeta functions of higher Heisenberg groups

Christopher Voll; Jianhao Shen

arxiv: 2605.23003 · v1 · pith:M4EDOH43new · submitted 2026-05-21 · 🧮 math.GR · math.CO· math.RA

Symplectic lattice counting and zeta functions of higher Heisenberg groups

Jianhao Shen , Christopher Voll This is my paper

Pith reviewed 2026-05-25 05:14 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.RA

keywords subalgebra zeta functionshigher Heisenberg Lie algebrassymplectic formsHecke theorydiscrete valuation ringslattice enumerationnilpotent Lie algebrassubgroup growth

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The pith

Explicit formulae are derived for the subalgebra zeta functions of all higher Heisenberg Lie algebras over any compact discrete valuation ring.

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

The paper develops Hecke-theoretic techniques to count sublattices of a finite-rank o-lattice carrying a non-degenerate symplectic form, enumerated by two distinct invariants. These techniques are applied to produce explicit formulae for the subalgebra zeta functions of every higher Heisenberg Lie algebra over an arbitrary compact discrete valuation ring o. A sympathetic reader would care because the zeta functions record the distribution of subalgebras and thereby control subgroup growth and representation counts for these nilpotent groups; closed formulae turn previously inaccessible local data into concrete expressions. The work shows that the symplectic pairing supplies enough structure for the enumeration to close in every case.

Core claim

By developing Hecke-theoretic techniques for the enumeration, by two distinct invariants, of sublattices of an o-lattice of finite rank endowed with a non-degenerate symplectic form, the authors derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring o.

What carries the argument

Hecke-theoretic enumeration techniques for counting sublattices by two invariants of a non-degenerate symplectic form on an o-lattice of finite rank

Load-bearing premise

The Hecke-theoretic enumeration techniques developed for counting sublattices by two invariants of a non-degenerate symplectic form on an o-lattice of finite rank are sufficient to produce the claimed explicit formulae for every higher Heisenberg Lie algebra.

What would settle it

Compute the subalgebra zeta function directly for the 5-dimensional higher Heisenberg Lie algebra over Z_p for a small prime p and verify whether the resulting rational function matches the explicit formula given by the counting method.

read the original abstract

We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by two distinct invariants, of sublattices of an $\mathfrak{o}$-lattice of finite rank endowed with a non-degenerate symplectic form.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shen and Voll give explicit formulas for subalgebra zeta functions of all higher Heisenberg Lie algebras over compact DVRs via new Hecke methods for two-invariant symplectic sublattice counts.

read the letter

The main takeaway is that this paper delivers explicit formulas for the subalgebra zeta functions of the full family of higher Heisenberg Lie algebras over any compact discrete valuation ring. The key technical step is a Hecke-theoretic setup that enumerates sublattices by two invariants at once while respecting a non-degenerate symplectic form on the underlying o-lattice. That approach appears tailored to capture the bracket-closed condition without extra fitting or post-hoc adjustments. If the derivations hold, it supplies concrete data that can serve as test cases for broader questions about growth and functional equations in this area. The abstract and stress-test description give no indication of internal contradictions or hidden rank restrictions, so the central claim looks consistent on its face. The Hecke machinery is presented as new for this simultaneous two-invariant counting task, which is a modest but genuine addition to the existing lattice-enumeration toolkit used in group zeta-function work. One soft spot is that the formulas themselves are not visible here, so it is impossible to check simplification, pole structure, or edge cases for higher ranks directly. That is a normal limitation when only the abstract is in view, not a sign of a deeper problem. The paper is written for specialists already working on zeta functions of nilpotent Lie algebras or p-adic groups. Anyone who needs explicit examples or wants to see how Hecke methods extend to symplectic settings will find it useful. It is narrow enough that it will not interest a broad audience, but the explicitness and the new counting technique are enough to justify sending it to referees rather than desk-rejecting. I would recommend peer review.

Referee Report

0 major / 0 minor

Summary. The manuscript claims to derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring o. This is achieved by developing Hecke-theoretic techniques for the enumeration, by two distinct invariants, of sublattices of an o-lattice of finite rank endowed with a non-degenerate symplectic form.

Significance. If the explicit formulae and the supporting enumeration techniques can be verified, the work would advance the computation of subalgebra zeta functions for nilpotent Lie algebras over p-adic rings, building on prior Hecke-algebra methods for lattice counting. The two-invariant symplectic enumeration approach, if parameter-free and applicable uniformly across ranks, could extend to other classes of Lie algebras or groups with bilinear forms.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments were listed in the report, so we have no points to address point-by-point at this stage. We remain available to provide additional verification or clarification of the explicit formulae or the Hecke-theoretic techniques if requested.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states it derives explicit formulae for subalgebra zeta functions by developing Hecke-theoretic enumeration techniques for sublattices counted by two invariants under a non-degenerate symplectic form on an o-lattice. This approach introduces new counting methods to handle the bracket-closed subalgebra condition for higher Heisenberg Lie algebras, without reducing any claimed prediction or formula to a fitted parameter, self-definition, or load-bearing self-citation chain. The central derivation chain is independent of its target outputs by construction, as the techniques are presented as external to the final zeta function expressions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5571 in / 1024 out tokens · 14935 ms · 2026-05-25T05:14:05.150395+00:00 · methodology

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Works this paper leans on

34 extracted references · 34 canonical work pages

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Alfes, J

C. Alfes, J. Maglione, and C. Voll.Symplectic Hecke eigenbases from Ehrhart polyno- mials. 2025. arXiv:2507.11728 [math.CO].url:https://arxiv.org/abs/2507. 11728

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M. N. Berman, I. Glazer, and M. M. Schein. “Pro-isomorphic zeta functions of nilpo- tent groups and Lie rings under base extension”. In:Trans. Amer. Math. Soc.375.2 (2022), pp. 1051–1100

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Proisomorphic zeta functions of groups

M. N. Berman. “Proisomorphic zeta functions of groups”. Doctoral dissertation. Ox- ford: University of Oxford, 2005. REFERENCES 49

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Blomer and C

V. Blomer and C. Voll.Analytic properties of representation zeta functions of groups of typeA 2. to appear in Math. Ann. 2026. arXiv:2509 . 13254 [math.NT].url: https://arxiv.org/abs/2509.13254

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Generalized Igusa functions and ideal growth in nilpotent Lie rings

A. Carnevale, M. M. Schein, and C. Voll. “Generalized Igusa functions and ideal growth in nilpotent Lie rings”. In:Algebra Number Theory18.3 (2024), pp. 537–582. doi:10.2140/ant.2024.18.537

work page doi:10.2140/ant.2024.18.537 2024
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Enumerating traceless matrices over compact discrete valuation rings

A. Carnevale, S. Shechter, and C. Voll. “Enumerating traceless matrices over compact discrete valuation rings”. In:Israel J. Math.227.2 (2018), pp. 957–986

work page 2018
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Reduced zeta functions of Lie algebras

A. Evseev. “Reduced zeta functions of Lie algebras”. In:J. Reine Angew. Math.633 (2009), pp. 197–211.doi:10.1515/CRELLE.2009.065

work page doi:10.1515/crelle.2009.065 2009
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Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field

S. Ezzat. “Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field”. In:J. Algebra397 (2014), pp. 609–624

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Gasper and M

G. Gasper and M. Rahman.Basic Hypergeometric Series. 2nd ed. Vol. 96. Encyclo- pedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 2004.isbn: 9780521833578.doi:10.1017/CBO9780511526251

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Subgroups of finite index in nilpotent groups

F. J. Grunewald, D. Segal, and G. C. Smith. “Subgroups of finite index in nilpotent groups”. In:Invent. Math.93.1 (1988), pp. 185–223.doi:10.1007/BF01393692

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Spherical functions and local densities of alternating forms

Y. Hironaka and F. Sato. “Spherical functions and local densities of alternating forms”. In:Amer. J. Math.110.3 (1988), pp. 473–512.doi:10.2307/2374620

work page doi:10.2307/2374620 1988
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Universalp-adic zeta functions and their functional equations

J.-i. Igusa. “Universalp-adic zeta functions and their functional equations”. In:Amer. J. Math.111.5 (1989), pp. 671–716

work page 1989
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Igusa-type functions associated to finite formed spaces and their functional equations

B. Klopsch and C. Voll. “Igusa-type functions associated to finite formed spaces and their functional equations”. In:Trans. Amer. Math. Soc.361.8 (2009), pp. 4405–4436. doi:10.1090/S0002-9947-09-04671-6

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P. Luschny.OEIS Sequence A393141. Accessed: 2026-04-27. 2026.url:https:// oeis.org/A393141

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I. G. Macdonald.Symmetric Functions and Hall Polynomials. 2nd ed. Oxford Math- ematical Monographs. Oxford: Oxford University Press, 1995.isbn: 9780198534891

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Maglione and C

J. Maglione and C. Voll.Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration. 2025. arXiv:2410 . 08075 [math.CO].url:https : / / arxiv . org/abs/2410.08075

work page arXiv 2025
[18]

Signed permutation statistics

V. Reiner. “Signed permutation statistics”. In:European J. Combin.14.6 (1993), pp. 553–567

work page 1993
[19]

Computing local zeta functions of groups, algebras, and modules

T. Rossmann. “Computing local zeta functions of groups, algebras, and modules”. In:Trans. Amer. Math. Soc.370.7 (2018), pp. 4841–4879

work page 2018
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Rossmann.Zeta, version 0.4.2

T. Rossmann.Zeta, version 0.4.2. Seehttps : / / github . com / torossmann / Zeta. 2022

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Analytic properties of zeta functions and sub- group growth

M. du Sautoy and F. Grunewald. “Analytic properties of zeta functions and sub- group growth”. In:Annals of Mathematics152.3 (2000), pp. 793–833.doi:10.2307/ 2661355. 50 REFERENCES

work page 2000
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du Sautoy and L

M. du Sautoy and L. Woodward.Zeta Functions of Groups and Rings. Vol. 1925. Lecture Notes in Mathematics. Berlin: Springer, 2008, pp. xii+208.isbn: 978-3-540- 74701-7.doi:10.1007/978-3-540-74776-5

work page doi:10.1007/978-3-540-74776-5 1925
[23]

Normal zeta functions of the Heisenberg groups over number rings I: the unramified case

M. M. Schein and C. Voll. “Normal zeta functions of the Heisenberg groups over number rings I: the unramified case”. In:J. Lond. Math. Soc. (2)91.1 (2015), pp. 19– 46.doi:10.1112/jlms/jdu061

work page doi:10.1112/jlms/jdu061 2015
[24]

Normal zeta functions of the Heisenberg groups over number rings II—the non-split case

M. M. Schein and C. Voll. “Normal zeta functions of the Heisenberg groups over number rings II—the non-split case”. In:Israel J. Math.211.1 (2016), pp. 171–195. doi:10.1007/s11856-015-1271-8

work page doi:10.1007/s11856-015-1271-8 2016
[25]

Groups with pairings, Hall modules, and Hall–Littlewood polynomials

J. Shen and R. Van Peski. “Groups with pairings, Hall modules, and Hall–Littlewood polynomials”. In:Int. Math. Res. Not. IMRN2025.22 (2025), rnaf339.doi:10.1093/ imrn/rnaf339

work page 2025
[26]

Shen and R

J. Shen and R. Van Peski.Non-Archimedean GUE corners and Hecke modules. 2024. arXiv:2412.05999 [math.PR].url:https://arxiv.org/abs/2412.05999

work page arXiv 2024
[27]

Siconolfi, M

V. Siconolfi, M. Vantomme, and C. Voll.Subgroup growth in free class-2-nilpotent groups. to appear in J. Combin. Algebra. 2024. arXiv:2305.19665 [math.GR].url: https://arxiv.org/abs/2305.19665

work page arXiv 2024
[28]

R. P. Stanley.Enumerative Combinatorics. Volume 1. 2nd ed. Vol. 49. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2011. doi:10.1017/CBO9781139058520

work page doi:10.1017/cbo9781139058520 2011
[29]

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

A. Stasinski and C. Voll. “Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B”. In:Amer. J. Math.136.2 (2014), pp. 501–550.doi:10.1353/ajm.2014.0010

work page doi:10.1353/ajm.2014.0010 2014
[30]

Counting subgroups in a family of nilpotent semi-direct products

C. Voll. “Counting subgroups in a family of nilpotent semi-direct products”. In:Bull. Lond. Math. Soc.38.5 (2006), pp. 743–752.doi:10.1112/S0024609306018881

work page doi:10.1112/s0024609306018881 2006
[31]

Functional equations for local normal zeta functions of nilpotent groups

C. Voll. “Functional equations for local normal zeta functions of nilpotent groups”. In:Geom. Funct. Anal.15.1 (2005). With an appendix by Arnaud Beauville, pp. 274– 295.doi:10.1007/s00039-005-0506-y

work page doi:10.1007/s00039-005-0506-y 2005
[32]

Functional equations for zeta functions of groups and rings

C. Voll. “Functional equations for zeta functions of groups and rings”. In:Ann. of Math. (2)172.2 (2010), pp. 1181–1218.doi:10.4007/annals.2010.172.1181

work page doi:10.4007/annals.2010.172.1181 2010
[33]

Partitions, flags, tableaux: combinatorial aspects of lattice enumeration

C. Voll. “Partitions, flags, tableaux: combinatorial aspects of lattice enumeration”. In:Nieuw Arch. Wiskd. (5)26.2 (2025), pp. 74–77

work page 2025
[34]

Zeta functions of groups: computer calculations and functional equa- tions

L. Woodward. “Zeta functions of groups: computer calculations and functional equa- tions”. Doctoral dissertation. Oxford: University of Oxford, 2005. Email address:jshen@math.uni-bielefeld.de Email address:C.Voll.98@cantab.net F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, D-33501 Bielefeld, Germany

work page 2005

[1] [1]

Alfes, J

C. Alfes, J. Maglione, and C. Voll.Symplectic Hecke eigenbases from Ehrhart polyno- mials. 2025. arXiv:2507.11728 [math.CO].url:https://arxiv.org/abs/2507. 11728

work page arXiv 2025

[2] [2]

Pro-isomorphic zeta functions of nilpo- tent groups and Lie rings under base extension

M. N. Berman, I. Glazer, and M. M. Schein. “Pro-isomorphic zeta functions of nilpo- tent groups and Lie rings under base extension”. In:Trans. Amer. Math. Soc.375.2 (2022), pp. 1051–1100

work page 2022

[3] [3]

Proisomorphic zeta functions of groups

M. N. Berman. “Proisomorphic zeta functions of groups”. Doctoral dissertation. Ox- ford: University of Oxford, 2005. REFERENCES 49

work page 2005

[4] [4]

Blomer and C

V. Blomer and C. Voll.Analytic properties of representation zeta functions of groups of typeA 2. to appear in Math. Ann. 2026. arXiv:2509 . 13254 [math.NT].url: https://arxiv.org/abs/2509.13254

work page arXiv 2026

[5] [5]

q-Eulerian polynomials arising from Coxeter groups

F. Brenti. “q-Eulerian polynomials arising from Coxeter groups”. In:European J. Combin.15.5 (1994), pp. 417–441.doi:10.1006/eujc.1994.1046

work page doi:10.1006/eujc.1994.1046 1994

[6] [6]

Generalized Igusa functions and ideal growth in nilpotent Lie rings

A. Carnevale, M. M. Schein, and C. Voll. “Generalized Igusa functions and ideal growth in nilpotent Lie rings”. In:Algebra Number Theory18.3 (2024), pp. 537–582. doi:10.2140/ant.2024.18.537

work page doi:10.2140/ant.2024.18.537 2024

[7] [7]

Enumerating traceless matrices over compact discrete valuation rings

A. Carnevale, S. Shechter, and C. Voll. “Enumerating traceless matrices over compact discrete valuation rings”. In:Israel J. Math.227.2 (2018), pp. 957–986

work page 2018

[8] [8]

Reduced zeta functions of Lie algebras

A. Evseev. “Reduced zeta functions of Lie algebras”. In:J. Reine Angew. Math.633 (2009), pp. 197–211.doi:10.1515/CRELLE.2009.065

work page doi:10.1515/crelle.2009.065 2009

[9] [9]

Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field

S. Ezzat. “Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field”. In:J. Algebra397 (2014), pp. 609–624

work page 2014

[10] [10]

Gasper and M

G. Gasper and M. Rahman.Basic Hypergeometric Series. 2nd ed. Vol. 96. Encyclo- pedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 2004.isbn: 9780521833578.doi:10.1017/CBO9780511526251

work page doi:10.1017/cbo9780511526251 2004

[11] [11]

Subgroups of finite index in nilpotent groups

F. J. Grunewald, D. Segal, and G. C. Smith. “Subgroups of finite index in nilpotent groups”. In:Invent. Math.93.1 (1988), pp. 185–223.doi:10.1007/BF01393692

work page doi:10.1007/bf01393692 1988

[12] [12]

Spherical functions and local densities of alternating forms

Y. Hironaka and F. Sato. “Spherical functions and local densities of alternating forms”. In:Amer. J. Math.110.3 (1988), pp. 473–512.doi:10.2307/2374620

work page doi:10.2307/2374620 1988

[13] [13]

Universalp-adic zeta functions and their functional equations

J.-i. Igusa. “Universalp-adic zeta functions and their functional equations”. In:Amer. J. Math.111.5 (1989), pp. 671–716

work page 1989

[14] [14]

Igusa-type functions associated to finite formed spaces and their functional equations

B. Klopsch and C. Voll. “Igusa-type functions associated to finite formed spaces and their functional equations”. In:Trans. Amer. Math. Soc.361.8 (2009), pp. 4405–4436. doi:10.1090/S0002-9947-09-04671-6

work page doi:10.1090/s0002-9947-09-04671-6 2009

[15] [15]

Luschny.OEIS Sequence A393141

P. Luschny.OEIS Sequence A393141. Accessed: 2026-04-27. 2026.url:https:// oeis.org/A393141

work page 2026

[16] [16]

I. G. Macdonald.Symmetric Functions and Hall Polynomials. 2nd ed. Oxford Math- ematical Monographs. Oxford: Oxford University Press, 1995.isbn: 9780198534891

work page 1995

[17] [17]

Maglione and C

J. Maglione and C. Voll.Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration. 2025. arXiv:2410 . 08075 [math.CO].url:https : / / arxiv . org/abs/2410.08075

work page arXiv 2025

[18] [18]

Signed permutation statistics

V. Reiner. “Signed permutation statistics”. In:European J. Combin.14.6 (1993), pp. 553–567

work page 1993

[19] [19]

Computing local zeta functions of groups, algebras, and modules

T. Rossmann. “Computing local zeta functions of groups, algebras, and modules”. In:Trans. Amer. Math. Soc.370.7 (2018), pp. 4841–4879

work page 2018

[20] [20]

Rossmann.Zeta, version 0.4.2

T. Rossmann.Zeta, version 0.4.2. Seehttps : / / github . com / torossmann / Zeta. 2022

work page 2022

[21] [21]

Analytic properties of zeta functions and sub- group growth

M. du Sautoy and F. Grunewald. “Analytic properties of zeta functions and sub- group growth”. In:Annals of Mathematics152.3 (2000), pp. 793–833.doi:10.2307/ 2661355. 50 REFERENCES

work page 2000

[22] [22]

du Sautoy and L

M. du Sautoy and L. Woodward.Zeta Functions of Groups and Rings. Vol. 1925. Lecture Notes in Mathematics. Berlin: Springer, 2008, pp. xii+208.isbn: 978-3-540- 74701-7.doi:10.1007/978-3-540-74776-5

work page doi:10.1007/978-3-540-74776-5 1925

[23] [23]

Normal zeta functions of the Heisenberg groups over number rings I: the unramified case

M. M. Schein and C. Voll. “Normal zeta functions of the Heisenberg groups over number rings I: the unramified case”. In:J. Lond. Math. Soc. (2)91.1 (2015), pp. 19– 46.doi:10.1112/jlms/jdu061

work page doi:10.1112/jlms/jdu061 2015

[24] [24]

Normal zeta functions of the Heisenberg groups over number rings II—the non-split case

M. M. Schein and C. Voll. “Normal zeta functions of the Heisenberg groups over number rings II—the non-split case”. In:Israel J. Math.211.1 (2016), pp. 171–195. doi:10.1007/s11856-015-1271-8

work page doi:10.1007/s11856-015-1271-8 2016

[25] [25]

Groups with pairings, Hall modules, and Hall–Littlewood polynomials

J. Shen and R. Van Peski. “Groups with pairings, Hall modules, and Hall–Littlewood polynomials”. In:Int. Math. Res. Not. IMRN2025.22 (2025), rnaf339.doi:10.1093/ imrn/rnaf339

work page 2025

[26] [26]

Shen and R

J. Shen and R. Van Peski.Non-Archimedean GUE corners and Hecke modules. 2024. arXiv:2412.05999 [math.PR].url:https://arxiv.org/abs/2412.05999

work page arXiv 2024

[27] [27]

Siconolfi, M

V. Siconolfi, M. Vantomme, and C. Voll.Subgroup growth in free class-2-nilpotent groups. to appear in J. Combin. Algebra. 2024. arXiv:2305.19665 [math.GR].url: https://arxiv.org/abs/2305.19665

work page arXiv 2024

[28] [28]

R. P. Stanley.Enumerative Combinatorics. Volume 1. 2nd ed. Vol. 49. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2011. doi:10.1017/CBO9781139058520

work page doi:10.1017/cbo9781139058520 2011

[29] [29]

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

A. Stasinski and C. Voll. “Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B”. In:Amer. J. Math.136.2 (2014), pp. 501–550.doi:10.1353/ajm.2014.0010

work page doi:10.1353/ajm.2014.0010 2014

[30] [30]

Counting subgroups in a family of nilpotent semi-direct products

C. Voll. “Counting subgroups in a family of nilpotent semi-direct products”. In:Bull. Lond. Math. Soc.38.5 (2006), pp. 743–752.doi:10.1112/S0024609306018881

work page doi:10.1112/s0024609306018881 2006

[31] [31]

Functional equations for local normal zeta functions of nilpotent groups

C. Voll. “Functional equations for local normal zeta functions of nilpotent groups”. In:Geom. Funct. Anal.15.1 (2005). With an appendix by Arnaud Beauville, pp. 274– 295.doi:10.1007/s00039-005-0506-y

work page doi:10.1007/s00039-005-0506-y 2005

[32] [32]

Functional equations for zeta functions of groups and rings

C. Voll. “Functional equations for zeta functions of groups and rings”. In:Ann. of Math. (2)172.2 (2010), pp. 1181–1218.doi:10.4007/annals.2010.172.1181

work page doi:10.4007/annals.2010.172.1181 2010

[33] [33]

Partitions, flags, tableaux: combinatorial aspects of lattice enumeration

C. Voll. “Partitions, flags, tableaux: combinatorial aspects of lattice enumeration”. In:Nieuw Arch. Wiskd. (5)26.2 (2025), pp. 74–77

work page 2025

[34] [34]

Zeta functions of groups: computer calculations and functional equa- tions

L. Woodward. “Zeta functions of groups: computer calculations and functional equa- tions”. Doctoral dissertation. Oxford: University of Oxford, 2005. Email address:jshen@math.uni-bielefeld.de Email address:C.Voll.98@cantab.net F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, D-33501 Bielefeld, Germany

work page 2005