When will the Stanley depth increase?

Shen, Yi-Huang

doi:10.1090/S0002-9939-2013-12003-4

When will the Stanley depth increase?

Author: Yi-Huang Shen
Journal: Proc. Amer. Math. Soc. 141 (2013), 2265-2274
MSC (2010): Primary 05E45, 05E40, 06A07; Secondary 13C13, 05C70
DOI: https://doi.org/10.1090/S0002-9939-2013-12003-4
Published electronically: March 20, 2013
MathSciNet review: 3043008
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Abstract: Let $I\subset S=\mathbb{K},[x_1,\dots ,x_n]$ be an ideal generated by squarefree monomials of degree $\ge d$ . If the number of degree minimal generating monomials is $\mu _d(I)\le \min (\binom {n}{d+1},\sum _{j=1}^{n-d}\binom {2j-1}{j})$ , then the Stanley depth $\operatorname {sdepth}_S(I)\ge d+1$ .

[BEOS09] Vittoria Bonanzinga, Viviana Ene, Anda Olteanu, and Loredana Sorrenti, An overview on the minimal free resolutions of lexsegment ideals, Combinatorial aspects of commutative algebra, Contemp. Math., vol. 502, Amer. Math. Soc., Providence, RI, 2009, pp. 5–24. MR 2583270, https://doi.org/10.1090/conm/502/09853
[BH93] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
[BHK$^+$10] Csaba Biró, David M. Howard, Mitchel T. Keller, William T. Trotter, and Stephen J. Young, Interval partitions and Stanley depth, J. Combin. Theory Ser. A 117 (2010), no. 4, 475–482. MR 2592896, https://doi.org/10.1016/j.jcta.2009.07.008
[BKU10] Winfried Bruns, Christian Krattenthaler, and Jan Uliczka, Stanley decompositions and Hilbert depth in the Koszul complex, J. Commut. Algebra 2 (2010), no. 3, 327–357. MR 2728147, https://doi.org/10.1216/JCA-2010-2-3-327
[Cim09] Mircea Cimpoeaş, Stanley depth of square free Veronese ideals (2009), available at arXiv:0907.1232.
[Duv94] Art M. Duval, A combinatorial decomposition of simplicial complexes, Israel J. Math. 87 (1994), no. 1-3, 77–87. MR 1286816, https://doi.org/10.1007/BF02772984
[Gar80] Adriano M. Garsia, Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math. 38 (1980), no. 3, 229–266. MR 597728, https://doi.org/10.1016/0001-8708(80)90006-7
[GK78] Curtis Greene and Daniel J. Kleitman, Proof techniques in the theory of finite sets, Studies in combinatorics, MAA Stud. Math., vol. 17, Math. Assoc. America, Washington, D.C., 1978, pp. 22–79. MR 513002
[HJY08] Jürgen Herzog, Ali Soleyman Jahan, and Siamak Yassemi, Stanley decompositions and partitionable simplicial complexes, J. Algebraic Combin. 27 (2008), no. 1, 113–125. MR 2366164, https://doi.org/10.1007/s10801-007-0076-1
[HVZ09] Jürgen Herzog, Marius Vladoiu, and Xinxian Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra 322 (2009), no. 9, 3151–3169. MR 2567414, https://doi.org/10.1016/j.jalgebra.2008.01.006
[KSSY11] Mitchel T. Keller, Yi-Huang Shen, Noah Streib, and Stephen J. Young, On the Stanley depth of squarefree Veronese ideals, J. Algebraic Combin. 33 (2011), no. 2, 313–324. MR 2765327, https://doi.org/10.1007/s10801-010-0249-1
[KY09] Mitchel T. Keller and Stephen J. Young, Stanley depth of squarefree monomial ideals, J. Algebra 322 (2009), no. 10, 3789–3792. MR 2568363, https://doi.org/10.1016/j.jalgebra.2009.05.021
[Oka11] Ryota Okazaki, A lower bound of Stanley depth of monomial ideals, J. Commut. Algebra 3 (2011), no. 1, 83–88. MR 2782700, https://doi.org/10.1216/JCA-2011-3-1-83
[She09] Yi Huang Shen, Stanley depth of complete intersection monomial ideals and upper-discrete partitions, J. Algebra 321 (2009), no. 4, 1285–1292. MR 2489900, https://doi.org/10.1016/j.jalgebra.2008.11.010
[Sta79] Richard P. Stanley, Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249 (1979), no. 1, 139–157. MR 526314, https://doi.org/10.1090/S0002-9947-1979-0526314-6
[Sta82] Richard P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. MR 666158, https://doi.org/10.1007/BF01394054
[vLW01] J. H. van Lint and R. M. Wilson, A course in combinatorics, 2nd ed., Cambridge University Press, Cambridge, 2001. MR 1871828