Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes
Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Cisto, Carmelo [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Anmerkung: |
© The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Bulletin of the Iranian Mathematical Society - Singapore : Springer Singapore, 2001, 49(2023), 3 vom: 03. Apr. |
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| Übergeordnetes Werk: |
volume:49 ; year:2023 ; number:3 ; day:03 ; month:04 |
| Links: |
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| DOI / URN: |
10.1007/s41980-023-00769-5 |
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| Katalog-ID: |
SPR049938959 |
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| 520 | |a Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. | ||
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| 650 | 4 | |a Hilbert–Poincaré series |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Rook polynomial |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Gorenstein |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Navarra, Francesco |0 (orcid)0000-0002-4914-158X |4 aut | |
| 700 | 1 | |a Utano, Rosanna |4 aut | |
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10.1007/s41980-023-00769-5 doi (DE-627)SPR049938959 (SPR)s41980-023-00769-5-e DE-627 ger DE-627 rakwb eng Cisto, Carmelo verfasserin aut Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. Polyominoes (dpeaa)DE-He213 Hilbert–Poincaré series (dpeaa)DE-He213 Rook polynomial (dpeaa)DE-He213 Gorenstein (dpeaa)DE-He213 Navarra, Francesco (orcid)0000-0002-4914-158X aut Utano, Rosanna aut Enthalten in Bulletin of the Iranian Mathematical Society Singapore : Springer Singapore, 2001 49(2023), 3 vom: 03. Apr. (DE-627)573093903 (DE-600)2440200-X 1735-8515 nnns volume:49 year:2023 number:3 day:03 month:04 https://dx.doi.org/10.1007/s41980-023-00769-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 49 2023 3 03 04 |
| spelling |
10.1007/s41980-023-00769-5 doi (DE-627)SPR049938959 (SPR)s41980-023-00769-5-e DE-627 ger DE-627 rakwb eng Cisto, Carmelo verfasserin aut Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. Polyominoes (dpeaa)DE-He213 Hilbert–Poincaré series (dpeaa)DE-He213 Rook polynomial (dpeaa)DE-He213 Gorenstein (dpeaa)DE-He213 Navarra, Francesco (orcid)0000-0002-4914-158X aut Utano, Rosanna aut Enthalten in Bulletin of the Iranian Mathematical Society Singapore : Springer Singapore, 2001 49(2023), 3 vom: 03. Apr. (DE-627)573093903 (DE-600)2440200-X 1735-8515 nnns volume:49 year:2023 number:3 day:03 month:04 https://dx.doi.org/10.1007/s41980-023-00769-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 49 2023 3 03 04 |
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10.1007/s41980-023-00769-5 doi (DE-627)SPR049938959 (SPR)s41980-023-00769-5-e DE-627 ger DE-627 rakwb eng Cisto, Carmelo verfasserin aut Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. Polyominoes (dpeaa)DE-He213 Hilbert–Poincaré series (dpeaa)DE-He213 Rook polynomial (dpeaa)DE-He213 Gorenstein (dpeaa)DE-He213 Navarra, Francesco (orcid)0000-0002-4914-158X aut Utano, Rosanna aut Enthalten in Bulletin of the Iranian Mathematical Society Singapore : Springer Singapore, 2001 49(2023), 3 vom: 03. Apr. (DE-627)573093903 (DE-600)2440200-X 1735-8515 nnns volume:49 year:2023 number:3 day:03 month:04 https://dx.doi.org/10.1007/s41980-023-00769-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 49 2023 3 03 04 |
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10.1007/s41980-023-00769-5 doi (DE-627)SPR049938959 (SPR)s41980-023-00769-5-e DE-627 ger DE-627 rakwb eng Cisto, Carmelo verfasserin aut Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. Polyominoes (dpeaa)DE-He213 Hilbert–Poincaré series (dpeaa)DE-He213 Rook polynomial (dpeaa)DE-He213 Gorenstein (dpeaa)DE-He213 Navarra, Francesco (orcid)0000-0002-4914-158X aut Utano, Rosanna aut Enthalten in Bulletin of the Iranian Mathematical Society Singapore : Springer Singapore, 2001 49(2023), 3 vom: 03. Apr. (DE-627)573093903 (DE-600)2440200-X 1735-8515 nnns volume:49 year:2023 number:3 day:03 month:04 https://dx.doi.org/10.1007/s41980-023-00769-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 49 2023 3 03 04 |
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10.1007/s41980-023-00769-5 doi (DE-627)SPR049938959 (SPR)s41980-023-00769-5-e DE-627 ger DE-627 rakwb eng Cisto, Carmelo verfasserin aut Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. Polyominoes (dpeaa)DE-He213 Hilbert–Poincaré series (dpeaa)DE-He213 Rook polynomial (dpeaa)DE-He213 Gorenstein (dpeaa)DE-He213 Navarra, Francesco (orcid)0000-0002-4914-158X aut Utano, Rosanna aut Enthalten in Bulletin of the Iranian Mathematical Society Singapore : Springer Singapore, 2001 49(2023), 3 vom: 03. Apr. (DE-627)573093903 (DE-600)2440200-X 1735-8515 nnns volume:49 year:2023 number:3 day:03 month:04 https://dx.doi.org/10.1007/s41980-023-00769-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_266 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 49 2023 3 03 04 |
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Polyominoes</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Hilbert–Poincaré series</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Rook polynomial</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Gorenstein</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Navarra, Francesco</subfield><subfield code="0">(orcid)0000-0002-4914-158X</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Utano, Rosanna</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Bulletin of the Iranian Mathematical Society</subfield><subfield code="d">Singapore : Springer Singapore, 2001</subfield><subfield code="g">49(2023), 3 vom: 03. 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hilbert–poincaré series and gorenstein property for some non-simple polyominoes |
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Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes |
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Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstract_unstemmed |
Abstract Let %$\mathcal {P}%$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper, we give a combinatorial interpretation of the h-polynomial of %$K[\mathcal {P}]%$, showing that it is the rook polynomial of %$\mathcal {P}%$. It is known by Rinaldo and Romeo (J Algebr Comb 54:607–624, 2021), that if %$\mathcal {P}%$ is a simple thin polyomino, then the h-polynomial is equal to the rook polynomial of %$\mathcal {P}%$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert–Poincaré series of the coordinate ring attached to a closed path %$\mathcal {P}%$ having no zig-zag walks, as a combination of the Hilbert–Poincaré series of convenient simple thin polyominoes. As a consequence, we prove that the Krull dimension is equal to %$\vert V(\mathcal {P})\vert -\textrm{rank}\, \mathcal {P}%$ and the regularity of %$K[\mathcal {P}]%$ is the rook number of %$\mathcal {P}%$. Finally, we characterize the Gorenstein prime closed paths, proving that %$K[\mathcal {P}]%$ is Gorenstein if and only if %$\mathcal {P}%$ consists of maximal blocks of length three. © The Author(s) under exclusive licence to Iranian Mathematical Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Hilbert–Poincaré Series and Gorenstein Property for Some Non-simple Polyominoes |
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Navarra, Francesco Utano, Rosanna |
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10.1007/s41980-023-00769-5 |
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| score |
7.401634 |