On trees with the same restricted U-polynomial and the Prouhet–Tarry–Escott problem
This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given k, non-...
Saved in:
| Published in: | Discrete mathematics Vol. 340; no. 6; pp. 1435 - 1441 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article Publication |
| Language: | English |
| Published: |
Elsevier B.V
01.06.2017
|
| Subjects: | |
| ISSN: | 0012-365X, 1872-681X |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given k, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism. |
|---|---|
| AbstractList | This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given kk, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism.
Peer Reviewed This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given k, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism. |
| Author | de Mier, Anna Aliste-Prieto, José Zamora, José |
| Author_xml | – sequence: 1 givenname: José surname: Aliste-Prieto fullname: Aliste-Prieto, José email: jose.aliste@unab.cl organization: Departamento de Matemáticas, Universidad Andres Bello, Republica 220, Santiago, Chile – sequence: 2 givenname: Anna surname: de Mier fullname: de Mier, Anna email: anna.de.mier@upc.edu organization: Departament de Matemàtiques, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain – sequence: 3 givenname: José surname: Zamora fullname: Zamora, José email: josezamora@unab.cl organization: Departamento de Matemáticas, Universidad Andres Bello, Republica 220, Santiago, Chile |
| BookMark | eNp9kEtqwzAQQEVJoUnaC3SlC9iV_JWhmxLSDwTSRUKzE_J4TBQcK0hKS3a9Q2_Yk9RuAoUuspgfzBuGNyKD1rRIyC1nIWc8u9uElXYQRl0fsiJkvLggQy7yKMgEXw3IkDEeBXGWrq7IyLkN6-YsFkPyNm-pt4iOfmi_pn6N1KktUovOWw0eK7oMdqY5tGarVUNVW_0uvVqzX6P__vxaKGsPXZ06MN7TnTVlg9trclmrxuHNqY7J8nG6mDwHs_nTy-RhFkDCIx-AgCQrkddYlqLskqrTPClQAXIoRC4gEpBygXmcxEXKONSFKrM6yaBSPBHxmPDjXXB7kBYBLSgvjdJ_Qx8RyyMZszgpekacGGucs1hL0F55bVpvlW4kZ7KXKjeylyp7qZIVspPaodE_dGf1VtnDeej-CGEn4l2jlQ40toCV7n70sjL6HP4DNDmXAw |
| CitedBy_id | crossref_primary_10_1016_j_disc_2021_112682 crossref_primary_10_1016_j_jcta_2021_105496 crossref_primary_10_1016_j_jcta_2021_105572 crossref_primary_10_1016_j_disc_2024_114096 crossref_primary_10_1016_j_aam_2024_102718 crossref_primary_10_1016_j_ejc_2020_103143 crossref_primary_10_1016_j_endm_2018_06_032 crossref_primary_10_1016_j_jctb_2019_05_006 crossref_primary_10_1016_j_jcta_2022_105608 crossref_primary_10_1112_blms_13144 crossref_primary_10_1016_j_disc_2020_112255 crossref_primary_10_1137_17M1144805 crossref_primary_10_1137_20M1380314 crossref_primary_10_1137_22M148046X |
| Cites_doi | 10.1016/j.disc.2013.10.016 10.1006/jctb.2000.1988 10.1006/aima.1995.1020 10.1090/S0002-9904-1948-09071-1 10.1016/0095-8956(81)90068-X 10.1007/s000260050008 10.5802/aif.1706 10.1016/j.jcta.2007.05.008 10.1016/S0012-365X(99)00363-5 10.1016/j.disc.2013.12.006 10.1080/00029890.1959.11989269 |
| ContentType | Journal Article Publication |
| Contributor | Universitat Politècnica de Catalunya. MD - Matemàtica Discreta Universitat Politècnica de Catalunya. Departament de Matemàtiques |
| Contributor_xml | – sequence: 1 fullname: Universitat Politècnica de Catalunya. Departament de Matemàtiques – sequence: 2 fullname: Universitat Politècnica de Catalunya. MD - Matemàtica Discreta |
| Copyright | 2016 Elsevier B.V. info:eu-repo/semantics/openAccess |
| Copyright_xml | – notice: 2016 Elsevier B.V. – notice: info:eu-repo/semantics/openAccess |
| DBID | AAYXX CITATION XX2 |
| DOI | 10.1016/j.disc.2016.09.019 |
| DatabaseName | CrossRef Recercat |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1872-681X |
| EndPage | 1441 |
| ExternalDocumentID | oai_recercat_cat_2072_303498 10_1016_j_disc_2016_09_019 S0012365X16303077 |
| GroupedDBID | --K --M -DZ -~X .DC .~1 0R~ 1B1 1RT 1~. 1~5 29G 4.4 41~ 457 4G. 5GY 5VS 6I. 6OB 6TJ 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AASFE AAXUO ABAOU ABEFU ABFNM ABJNI ABMAC ABTAH ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACRLP ADBBV ADEZE ADIYS ADMUD AEBSH AEKER AENEX AEXQZ AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AI. AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 FA8 FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA HVGLF HZ~ IHE IXB J1W KOM M26 M41 MHUIS MO0 MVM N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SSW SSZ T5K TN5 UPT VH1 WH7 WUQ XJT XOL XPP ZCG ZMT ZY4 ~G- 9DU AATTM AAXKI AAYWO AAYXX ABUFD ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO ADVLN ADXHL AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD XX2 |
| ID | FETCH-LOGICAL-c412t-c8c46be1febb8bebbaf5749eace1c9878c28c518e73439501cf9ab6f46cda1483 |
| ISICitedReferencesCount | 19 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000398751100032&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0012-365X |
| IngestDate | Fri Nov 07 13:42:09 EST 2025 Sat Nov 29 06:17:53 EST 2025 Tue Nov 18 22:03:05 EST 2025 Fri Feb 23 02:22:13 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 6 |
| Keywords | Prouhet–Tarry– Escott problem Graph polynomials Chromatic symmetric function U-polynomial |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c412t-c8c46be1febb8bebbaf5749eace1c9878c28c518e73439501cf9ab6f46cda1483 |
| OpenAccessLink | https://recercat.cat/handle/2072/303498 |
| PageCount | 7 |
| ParticipantIDs | csuc_recercat_oai_recercat_cat_2072_303498 crossref_citationtrail_10_1016_j_disc_2016_09_019 crossref_primary_10_1016_j_disc_2016_09_019 elsevier_sciencedirect_doi_10_1016_j_disc_2016_09_019 |
| PublicationCentury | 2000 |
| PublicationDate | 2017-06-01 |
| PublicationDateYYYYMMDD | 2017-06-01 |
| PublicationDate_xml | – month: 06 year: 2017 text: 2017-06-01 day: 01 |
| PublicationDecade | 2010 |
| PublicationTitle | Discrete mathematics |
| PublicationYear | 2017 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | K. Markström, From the ising and potts models to the general graph homomorphism polynomial, arXiv preprint Orellana, Scott (b11) 2014; 320 K. Russell, All trees on 25 or fewer vertices have different chromatic symmetric function, (2013). Sarmiento (b13) 2000; 4 Stanley (b16) 1999 Bollobás, Pebody, Riordan (b2) 2000; 80 Bollobás, Riordan (b3) 2000; 219 Borwein, Ingalls (b5) 1994; 40 Brylawski (b6) 1981; 30 . Wright (b17) 1948; 8 Wright (b18) 1959 Aliste-Prieto, Zamora (b1) 2014; 315 Noble, Welsh (b10) 1999; 49 Borwein (b4) 2002 M. Loebl, J.-S. Sereni, Potts partition function and isomorphisms of trees, arXiv preprint Martin, Morin, Wagner (b9) 2008; 115 I. Smith, Z. Smith, P. Tian, Symmetric chromatic polynomial of trees, arXiv preprint Stanley (b15) 1995; 111 Sarmiento (10.1016/j.disc.2016.09.019_b13) 2000; 4 Martin (10.1016/j.disc.2016.09.019_b9) 2008; 115 Bollobás (10.1016/j.disc.2016.09.019_b3) 2000; 219 Aliste-Prieto (10.1016/j.disc.2016.09.019_b1) 2014; 315 10.1016/j.disc.2016.09.019_b8 Brylawski (10.1016/j.disc.2016.09.019_b6) 1981; 30 Stanley (10.1016/j.disc.2016.09.019_b16) 1999 Noble (10.1016/j.disc.2016.09.019_b10) 1999; 49 Borwein (10.1016/j.disc.2016.09.019_b4) 2002 Borwein (10.1016/j.disc.2016.09.019_b5) 1994; 40 Wright (10.1016/j.disc.2016.09.019_b17) 1948; 8 Bollobás (10.1016/j.disc.2016.09.019_b2) 2000; 80 Orellana (10.1016/j.disc.2016.09.019_b11) 2014; 320 10.1016/j.disc.2016.09.019_b7 10.1016/j.disc.2016.09.019_b14 Wright (10.1016/j.disc.2016.09.019_b18) 1959 10.1016/j.disc.2016.09.019_b12 Stanley (10.1016/j.disc.2016.09.019_b15) 1995; 111 |
| References_xml | – year: 2002 ident: b4 publication-title: Computational Excursions in Analysis and Number Theory – volume: 40 start-page: 3 year: 1994 end-page: 27 ident: b5 article-title: The Prouhet-Tarry-Escott problem revisited publication-title: Enseign. Math. (2) – year: 1999 ident: b16 publication-title: Cambridge Studies in Advanced Mathematics, Vol. 62 – start-page: 199 year: 1959 end-page: 201 ident: b18 article-title: Prouhet’s 1851 solution of the Tarry-Escott problem of 1910 publication-title: Amer. Math. Monthly – volume: 111 start-page: 166 year: 1995 end-page: 194 ident: b15 article-title: A symmetric function generalization of the chromatic polynomial of a graph publication-title: Adv. Math. – volume: 115 start-page: 237 year: 2008 end-page: 253 ident: b9 article-title: On distinguishing trees by their chromatic symmetric functions publication-title: J. Combin. Theory Ser. A – volume: 30 start-page: 233 year: 1981 end-page: 246 ident: b6 article-title: Intersection theory for graphs publication-title: J. Combin. Theory Ser. B – reference: K. Russell, All trees on 25 or fewer vertices have different chromatic symmetric function, (2013). – reference: . – reference: I. Smith, Z. Smith, P. Tian, Symmetric chromatic polynomial of trees, arXiv preprint – volume: 4 start-page: 227 year: 2000 end-page: 236 ident: b13 article-title: The polychromate and a chord diagram polynomial publication-title: Ann. Comb. – volume: 80 start-page: 320 year: 2000 end-page: 345 ident: b2 article-title: Contraction–deletion invariants for graphs publication-title: J. Combin. Theory Ser. B – reference: M. Loebl, J.-S. Sereni, Potts partition function and isomorphisms of trees, arXiv preprint – volume: 219 start-page: 1 year: 2000 end-page: 7 ident: b3 article-title: Polychromatic polynomials publication-title: Discrete Math. – volume: 315 start-page: 158 year: 2014 end-page: 164 ident: b1 article-title: Proper caterpillars are distinguished by their chromatic symmetric function publication-title: Discrete Math. – volume: 320 start-page: 1 year: 2014 end-page: 14 ident: b11 article-title: Graphs with equal chromatic symmetric functions publication-title: Discrete Math. – reference: K. Markström, From the ising and potts models to the general graph homomorphism polynomial, arXiv preprint – volume: 49 start-page: 1057 year: 1999 end-page: 1087 ident: b10 article-title: A weighted graph polynomial from chromatic invariants of knots publication-title: Ann. Inst. Fourier (Grenoble) – volume: 8 start-page: 755 year: 1948 end-page: 757 ident: b17 article-title: Equal sums of like powers publication-title: Bull. Amer. Math. Soc – volume: 315 start-page: 158 year: 2014 ident: 10.1016/j.disc.2016.09.019_b1 article-title: Proper caterpillars are distinguished by their chromatic symmetric function publication-title: Discrete Math. doi: 10.1016/j.disc.2013.10.016 – volume: 80 start-page: 320 issue: 2 year: 2000 ident: 10.1016/j.disc.2016.09.019_b2 article-title: Contraction–deletion invariants for graphs publication-title: J. Combin. Theory Ser. B doi: 10.1006/jctb.2000.1988 – volume: 111 start-page: 166 issue: 1 year: 1995 ident: 10.1016/j.disc.2016.09.019_b15 article-title: A symmetric function generalization of the chromatic polynomial of a graph publication-title: Adv. Math. doi: 10.1006/aima.1995.1020 – volume: 40 start-page: 3 issue: 1–2 year: 1994 ident: 10.1016/j.disc.2016.09.019_b5 article-title: The Prouhet-Tarry-Escott problem revisited publication-title: Enseign. Math. (2) – ident: 10.1016/j.disc.2016.09.019_b8 – year: 2002 ident: 10.1016/j.disc.2016.09.019_b4 – volume: 8 start-page: 755 year: 1948 ident: 10.1016/j.disc.2016.09.019_b17 article-title: Equal sums of like powers publication-title: Bull. Amer. Math. Soc doi: 10.1090/S0002-9904-1948-09071-1 – ident: 10.1016/j.disc.2016.09.019_b7 – year: 1999 ident: 10.1016/j.disc.2016.09.019_b16 – volume: 30 start-page: 233 issue: 2 year: 1981 ident: 10.1016/j.disc.2016.09.019_b6 article-title: Intersection theory for graphs publication-title: J. Combin. Theory Ser. B doi: 10.1016/0095-8956(81)90068-X – volume: 4 start-page: 227 issue: 2 year: 2000 ident: 10.1016/j.disc.2016.09.019_b13 article-title: The polychromate and a chord diagram polynomial publication-title: Ann. Comb. doi: 10.1007/s000260050008 – volume: 49 start-page: 1057 issue: 3 year: 1999 ident: 10.1016/j.disc.2016.09.019_b10 article-title: A weighted graph polynomial from chromatic invariants of knots publication-title: Ann. Inst. Fourier (Grenoble) doi: 10.5802/aif.1706 – ident: 10.1016/j.disc.2016.09.019_b14 – volume: 115 start-page: 237 issue: 2 year: 2008 ident: 10.1016/j.disc.2016.09.019_b9 article-title: On distinguishing trees by their chromatic symmetric functions publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2007.05.008 – ident: 10.1016/j.disc.2016.09.019_b12 – volume: 219 start-page: 1 issue: 1 year: 2000 ident: 10.1016/j.disc.2016.09.019_b3 article-title: Polychromatic polynomials publication-title: Discrete Math. doi: 10.1016/S0012-365X(99)00363-5 – volume: 320 start-page: 1 year: 2014 ident: 10.1016/j.disc.2016.09.019_b11 article-title: Graphs with equal chromatic symmetric functions publication-title: Discrete Math. doi: 10.1016/j.disc.2013.12.006 – start-page: 199 year: 1959 ident: 10.1016/j.disc.2016.09.019_b18 article-title: Prouhet’s 1851 solution of the Tarry-Escott problem of 1910 publication-title: Amer. Math. Monthly doi: 10.1080/00029890.1959.11989269 |
| SSID | ssj0001638 |
| Score | 2.3330853 |
| Snippet | This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same... |
| SourceID | csuc crossref elsevier |
| SourceType | Open Access Repository Enrichment Source Index Database Publisher |
| StartPage | 1435 |
| SubjectTerms | 05 Combinatorics 05C Graph theory 05E Algebraic combinatorics [formula omitted]-polynomial Chromatic symmetric function Classificació AMS Combinatorial analysis Combinatòria Graph polynomials Matemàtica discreta Matemàtiques i estadística Polinomis Polynomials Prouhet–Tarry– Escott problem UU-polynomial Àrees temàtiques de la UPC |
| Title | On trees with the same restricted U-polynomial and the Prouhet–Tarry–Escott problem |
| URI | https://dx.doi.org/10.1016/j.disc.2016.09.019 https://recercat.cat/handle/2072/303498 |
| Volume | 340 |
| WOSCitedRecordID | wos000398751100032&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1872-681X dateEnd: 20180131 omitProxy: false ssIdentifier: ssj0001638 issn: 0012-365X databaseCode: AIEXJ dateStart: 19950120 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lj9MwELaWLgc4IJ6ivJQDJ1ZBeTm2jxUUARK7leiKiIvluI7oqk2rNF3t3vgPiD_IL2EmdtIsi1aAxKFuk9qN6vky89mZByHPaWCUAcPip8BW_UTHsZ-nWvtCB0powwRVs6bYBDs85FkmJnt739tYmNMFK0t-dibW_1XUcA6EjaGzfyHu7kfhBHwGoUMLYof2jwR_VKL3uHFha8grN2qJxVGwQodGgnnsr1eLc4xHdokCsNOkWm2_mLp1foinqqrOu6Pxpsnh4MrP9Bnt6zkoHmDeB8suAWxH00cLBJE_geV4vXLPG-yT-bbHDLTK3BXULsvORHxWy1WlLo9w2xMh27lR2T2zS3EzVg9jqsSUZtYKWdXLWeSnPMz6ujm2uZwcCPuaFnlez2rjuvC3FsFuTpy8xCBn9ORLm7S2Tk1fzLT90Sa0oxlwVNR97BrZjxgVfED2R-_G2fvOxCOJtSbe_g0XjWUdB3-90gXGM9Cbre4Rnx6Zmd4mt9wqxBtZ9Nwhe6a8S25-2EnwHvl0VHoNjjzEkQffeIgjb4cjr48jD3DUdHI4-vH1W4MgeLfY8Rx27pPjN-Ppq7e-q8Lh6ySMal9znaS5CQuT5zyHRhWUJQIMtgm14IzriGsacsNiILc0CHUhVJ4WSapnChbb8QMyKFeleUi8OMgFL1QUUD1LFM8VqP8iimexpkEOa9shCduZktqlqMdKKQvZ-iKeSJxdibMrAyFhdofkoBuztglaruz9AgUggU2YSqtaYnb17gBfUcAiGWPWJj4ktBWTdHzU8kwJ4LriIo_-cdxjcmN3Bz0hg7ramqfkuj6t55vqmcPgT2w2rxI |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+trees+with+the+same+restricted+U-polynomial+and+the+Prouhet%E2%80%93Tarry%E2%80%93Escott+problem&rft.jtitle=Discrete+mathematics&rft.au=Aliste-Prieto%2C+Jos%C3%A9&rft.au=de+Mier%2C+Anna&rft.au=Zamora%2C+Jos%C3%A9&rft.date=2017-06-01&rft.pub=Elsevier+B.V&rft.issn=0012-365X&rft.eissn=1872-681X&rft.volume=340&rft.issue=6&rft.spage=1435&rft.epage=1441&rft_id=info:doi/10.1016%2Fj.disc.2016.09.019&rft.externalDocID=S0012365X16303077 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0012-365X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0012-365X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0012-365X&client=summon |