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src/doc/en/reference/references/index.rst

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@@ -188,7 +188,7 @@ REFERENCES:
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.. [AHM2018] Edward Allen, Joshua Hallam, Sarah Mason, *Dual Immaculate
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Quasisymmetric Functions Expand Positively into Young Quasisymmetric
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Schur Functions*. :arxiv:`1606.03519`
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.. [AHMP2008] \J.-P. Aumasson, L. Henzen, W. Meier, and R. C-W Phan,
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*Sha-3 proposal blake*; in Submission to NIST, (2008).
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@@ -1036,7 +1036,7 @@ REFERENCES:
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.. [BKSTY06] \A. Buch, A. Kresch, M. Shimozono, H. Tamvakis, and A. Yong.
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*Stable Grothendieck polynomials and* `K`-*theoretic factor sequences*.
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Math. Ann. **340** Issue 2, (2008), pp. 359--382.
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:arxiv:`math/0601514v1`.
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:arxiv:`math/0601514v1`.
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.. [BK2009] \J. Brundan and A. Kleshchev.
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*Graded decomposition numbers for cyclotomic Hecke algebras*.
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Volume 20, Springer (2007)
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.. [But2010] Peter Butkovič, *Max-linear systems. Theory and algorithms.*
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Springer Monographs in Mathematics. London: Springer. xvii, 272 p. (2010).
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Springer Monographs in Mathematics. London: Springer. xvii, 272 p. (2010).
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.. [Buell89] Duncan A. Buell.
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*Binary Quadratic Forms: Classical Theory and Modern Computations.*
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.. [CM] Benoit Collins, Sho Matsumoto, *On some properties of
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orthogonal Weingarten functions*, :arxiv:`0903.5143`.
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.. [CM2000] Paula A.A.B. Carvalho and Ian M. Musson. *Down-up algebras and
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their representation theory*. J. Algebra. **228** no. 1, (2000),
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pp. 286-310. :doi:`10.1006/jabr.1999.8263`
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.. [EG1987] Paul Edelman, Curtis Greene.
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*Balanced Tableaux*.
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Advances in Mathematics 63 (1987), pp. 42-99.
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:doi:`10.1016/0001-8708(87)90063-6`
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:doi:`10.1016/0001-8708(87)90063-6`
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.. [EGNO2015] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik,
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*Tensor Categories*, AMS Mathematical Surveys and Monographs 205 (2015).
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und Geometrie*, Dissertation, University of
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Bayreuth, 2013.
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.. [FFN2025] Wenjie Fang, Éric Fusy, Philippe Nadeau,
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.. [FFN2025] Wenjie Fang, Éric Fusy, Philippe Nadeau,
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*Tamari intervals and blossoming trees*.
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Combinatorial Theory, Volume 5, Issue 1.
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:doi:`10.5070/C65165015`.
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.. [GHJV1994] \E. Gamma, R. Helm, R. Johnson, J. Vlissides, *Design
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Patterns: Elements of Reusable Object-Oriented
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Software*. Addison-Wesley (1994). ISBN 0-201-63361-2.
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.. [GHM2008] \S. Galbraith, M. Harrison, D. Mireles Morales,
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*Efficient hyperelliptic arithmetic using balanced representation for divisors*,
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Algorithmic Number Theory: 8th International Symposium, ANTS-VIII Banff, Canada, May 17-22, 2008 Proceedings 8.
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.. [GR2018v5sol] Darij Grinberg, Victor Reiner.
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*Hopf Algebras In Combinatorics*,
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:arxiv:`1409.8356v5`, available with solutions at
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https://arxiv.org/src/1409.8356v5/anc/HopfComb-v73-with-solutions.pdf
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https://arxiv.org/src/1409.8356v5/anc/HopfComb-v73-with-solutions.pdf
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.. [Gri2005] \G. Grigorov, Kato's Euler System and the Main Conjecture,
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Harvard Ph.D. Thesis (2005).
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matroid representations*, Discrete Applied Mathematics
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154 (2006), pp. 1210-1222.
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.. [HLMW2011] James Haglund, Kurt Luoto, Sarah Mason, and Stephanie
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van Willigenburg. *Quasisymmetric Schur functions.*
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Journal of Combinatorial Theory, Series A 118, no. 2
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(2011): 463-490. :arxiv:`0810.2489`
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.. [HLNT09] \F. Hivert, J.-G. Luque, J.-C. Novelli, J.-Y. Thibon,
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*The (1-E)-transform in combinatorial Hopf algebras*.
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:arxiv:`math/0912.0184v2`
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.. [Read2011] \N. Reading, *Noncrossing partitions and the shard intersection
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order*, J. Algebraic Combin., 33 (2011), 483-530.
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:doi:`10.1007/s10801-010-0255-3`
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.. [ReSt2020] Nathan Reading and Salvatore Stella, *An affine almost positive
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roots model*, J. Comb. Algebra Volume 4, Issue 1, 2020, pp. 1--59
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.. [TW1980] \A.D. Thomas and G.V. Wood, Group Tables (Exeter: Shiva
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Publishing, 1980)
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.. [TW2018] Vasu Tewari and Stephanie van
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Willigenburg. *Quasisymmetric and noncommutative skew
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Pieri rules*. Advances in Applied Mathematics 100
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(2018): pp. 101-121. :arxiv:`1512.04614`
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.. [TY1984] Robert Endre Tarjan, Mihalis Yannakakis. *Simple linear-time
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algorithms to test chordality of graphs, test acyclicity of
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hypergraphs, and selectively reduce acyclic hypergraphs*.

src/sage/combinat/growth.py

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Dual graded graphs for (skew) quasisymmetric Schur functions.
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These were introduced by Vasu Tewari and Stephanie Van
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Willigenburg in :arxiv:`1512.04614`.
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Willigenburg in [TW2018]_.
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Currently, the backward rule is not implemented. The forward
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rule seems to agree with Mason's insertion algorithm from Section
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6.1 of http://www.math.ubc.ca/~steph/papers/QS5.pdf.
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6.1 of [HLMW2011]_.
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The vertices of the dual graded graph are
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:class:`~sage.combinat.composition.Compositions`::

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