The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings
Let ( P , ≤ ) be a finite poset. Define the numbers a 1 , a 2 ,… (respectively, c 1 , c 2 ,…) so that a 1 + … + a k (respectively, c 1 + … + c k ) is the maximal number of elements of P which may be covered by k antichains (respectively, k chains.) Then the number e ( P ) of linear extensions of pos...
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| Veröffentlicht in: | Order (Dordrecht) 2021-07, Vol.38 (2), p.323-328 |
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| creator | Bochkov, I. A. Petrov, F. V. |
| description | Let
(
P
,
≤
)
be a finite poset. Define the numbers
a
1
,
a
2
,… (respectively,
c
1
,
c
2
,…) so that
a
1
+ … +
a
k
(respectively,
c
1
+ … +
c
k
) is the maximal number of elements of
P
which may be covered by
k
antichains (respectively,
k
chains.) Then the number
e
(
P
)
of linear extensions of poset
P
is not less than
∏
a
i
!
and not more than
n
!
/
∏
c
i
!
. A corollary: if
P
is partitioned onto disjoint antichains of sizes
b
1
,
b
2
,…, then
e
(
P
)
≥
∏
b
i
!
. |
| doi_str_mv | 10.1007/s11083-020-09542-3 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2543086124</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2543086124</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-65c5969fe3632d91c8aae4e42320a89a8f4903599471dc50d3a2290ee3ad5233</originalsourceid><addsrcrecordid>eNp9kE9LwzAYxoMoOKdfwFPAc_RN3qRtjrPMKQz1MLyG2KauwyUzaUW_vd0qePP08sDzh_dHyCWHaw6Q3yTOoUAGAhhoJQXDIzLhKhdMixyPyQR4lrMCtDwlZyltAAC1yibkebV29Db0vk60CZF2g3zst68u0tDQZeudjXT-1Tmf2uATfWktLde29dT6ms5811YHVYZPF1v_ls7JSWPfk7v4vVOyupuvynu2fFo8lLMlq5DrjmWqUjrTjcMMRa15VVjrpJMCBdhC26KRGlBpLXNeVwpqtEJocA5trQTilFyNtbsYPnqXOrMJffTDohFKIhQZF3JwidFVxZBSdI3ZxXZr47fhYPbgzAjODODMAZzZV-MYSrv9Ry7-Vf-T-gHN527E</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2543086124</pqid></control><display><type>article</type><title>The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings</title><source>SpringerLink Journals</source><creator>Bochkov, I. A. ; Petrov, F. V.</creator><creatorcontrib>Bochkov, I. A. ; Petrov, F. V.</creatorcontrib><description>Let
(
P
,
≤
)
be a finite poset. Define the numbers
a
1
,
a
2
,… (respectively,
c
1
,
c
2
,…) so that
a
1
+ … +
a
k
(respectively,
c
1
+ … +
c
k
) is the maximal number of elements of
P
which may be covered by
k
antichains (respectively,
k
chains.) Then the number
e
(
P
)
of linear extensions of poset
P
is not less than
∏
a
i
!
and not more than
n
!
/
∏
c
i
!
. A corollary: if
P
is partitioned onto disjoint antichains of sizes
b
1
,
b
2
,…, then
e
(
P
)
≥
∏
b
i
!
.</description><identifier>ISSN: 0167-8094</identifier><identifier>EISSN: 1572-9273</identifier><identifier>DOI: 10.1007/s11083-020-09542-3</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Algebra ; Chains ; Discrete Mathematics ; Lattices ; Mathematics ; Mathematics and Statistics ; Order ; Ordered Algebraic Structures ; Set theory</subject><ispartof>Order (Dordrecht), 2021-07, Vol.38 (2), p.323-328</ispartof><rights>Springer Nature B.V. 2020</rights><rights>Springer Nature B.V. 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-65c5969fe3632d91c8aae4e42320a89a8f4903599471dc50d3a2290ee3ad5233</citedby><cites>FETCH-LOGICAL-c319t-65c5969fe3632d91c8aae4e42320a89a8f4903599471dc50d3a2290ee3ad5233</cites><orcidid>0000-0003-1693-2745</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11083-020-09542-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11083-020-09542-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51298</link.rule.ids></links><search><creatorcontrib>Bochkov, I. A.</creatorcontrib><creatorcontrib>Petrov, F. V.</creatorcontrib><title>The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings</title><title>Order (Dordrecht)</title><addtitle>Order</addtitle><description>Let
(
P
,
≤
)
be a finite poset. Define the numbers
a
1
,
a
2
,… (respectively,
c
1
,
c
2
,…) so that
a
1
+ … +
a
k
(respectively,
c
1
+ … +
c
k
) is the maximal number of elements of
P
which may be covered by
k
antichains (respectively,
k
chains.) Then the number
e
(
P
)
of linear extensions of poset
P
is not less than
∏
a
i
!
and not more than
n
!
/
∏
c
i
!
. A corollary: if
P
is partitioned onto disjoint antichains of sizes
b
1
,
b
2
,…, then
e
(
P
)
≥
∏
b
i
!
.</description><subject>Algebra</subject><subject>Chains</subject><subject>Discrete Mathematics</subject><subject>Lattices</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Order</subject><subject>Ordered Algebraic Structures</subject><subject>Set theory</subject><issn>0167-8094</issn><issn>1572-9273</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LwzAYxoMoOKdfwFPAc_RN3qRtjrPMKQz1MLyG2KauwyUzaUW_vd0qePP08sDzh_dHyCWHaw6Q3yTOoUAGAhhoJQXDIzLhKhdMixyPyQR4lrMCtDwlZyltAAC1yibkebV29Db0vk60CZF2g3zst68u0tDQZeudjXT-1Tmf2uATfWktLde29dT6ms5811YHVYZPF1v_ls7JSWPfk7v4vVOyupuvynu2fFo8lLMlq5DrjmWqUjrTjcMMRa15VVjrpJMCBdhC26KRGlBpLXNeVwpqtEJocA5trQTilFyNtbsYPnqXOrMJffTDohFKIhQZF3JwidFVxZBSdI3ZxXZr47fhYPbgzAjODODMAZzZV-MYSrv9Ry7-Vf-T-gHN527E</recordid><startdate>20210701</startdate><enddate>20210701</enddate><creator>Bochkov, I. A.</creator><creator>Petrov, F. V.</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-1693-2745</orcidid></search><sort><creationdate>20210701</creationdate><title>The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings</title><author>Bochkov, I. A. ; Petrov, F. V.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-65c5969fe3632d91c8aae4e42320a89a8f4903599471dc50d3a2290ee3ad5233</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algebra</topic><topic>Chains</topic><topic>Discrete Mathematics</topic><topic>Lattices</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Order</topic><topic>Ordered Algebraic Structures</topic><topic>Set theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bochkov, I. A.</creatorcontrib><creatorcontrib>Petrov, F. V.</creatorcontrib><collection>CrossRef</collection><jtitle>Order (Dordrecht)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bochkov, I. A.</au><au>Petrov, F. V.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings</atitle><jtitle>Order (Dordrecht)</jtitle><stitle>Order</stitle><date>2021-07-01</date><risdate>2021</risdate><volume>38</volume><issue>2</issue><spage>323</spage><epage>328</epage><pages>323-328</pages><issn>0167-8094</issn><eissn>1572-9273</eissn><abstract>Let
(
P
,
≤
)
be a finite poset. Define the numbers
a
1
,
a
2
,… (respectively,
c
1
,
c
2
,…) so that
a
1
+ … +
a
k
(respectively,
c
1
+ … +
c
k
) is the maximal number of elements of
P
which may be covered by
k
antichains (respectively,
k
chains.) Then the number
e
(
P
)
of linear extensions of poset
P
is not less than
∏
a
i
!
and not more than
n
!
/
∏
c
i
!
. A corollary: if
P
is partitioned onto disjoint antichains of sizes
b
1
,
b
2
,…, then
e
(
P
)
≥
∏
b
i
!
.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s11083-020-09542-3</doi><tpages>6</tpages><orcidid>https://orcid.org/0000-0003-1693-2745</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0167-8094 |
| ispartof | Order (Dordrecht), 2021-07, Vol.38 (2), p.323-328 |
| issn | 0167-8094 1572-9273 |
| language | eng |
| recordid | cdi_proquest_journals_2543086124 |
| source | SpringerLink Journals |
| subjects | Algebra Chains Discrete Mathematics Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures Set theory |
| title | The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings |
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