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T(n,k) is the number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors.
(history; published version)

#15 by Bruno Berselli at Mon Mar 22 06:27:37 EDT 2021

STATUS

reviewed

approved

#14 by Joerg Arndt at Mon Mar 22 05:50:43 EDT 2021

STATUS

proposed

reviewed

#13 by Michel Marcus at Mon Mar 22 03:45:46 EDT 2021

STATUS

editing

proposed

#12 by Michel Marcus at Mon Mar 22 03:45:41 EDT 2021

COMMENTS

Table starts

....7....22.....50......95......161.......252.......372........525........715

...12....51....144.....325......636......1127......1856.......2889.......4300

...21...121....422....1121.....2507......4977......9052......15393......24817

...37...292...1268....3985....10213.....22736.....45648......84681.....147565

...65...704...3823...14288....42182....105813....235538.....478467.....904111

..114..1691..11472...50995...173606....491533...1215616....2710413....5567530

..200..4059..34350..181336...710976...2269938...6233356...15250675...34054592

..351..9749.102896..644721..2908797..10462235..31868448...85473225..207289059

..616.23422.308419.2294193.11911516..48259083.163014678..479101189.1261310492

.1081.56268.924532.8166441.48807427.222798408.834763824.2688814689.7684922749

EXAMPLE

Table starts:

....7....22.....50......95......161.......252.......372........525........715

...12....51....144.....325......636......1127......1856.......2889.......4300

...21...121....422....1121.....2507......4977......9052......15393......24817

...37...292...1268....3985....10213.....22736.....45648......84681.....147565

...65...704...3823...14288....42182....105813....235538.....478467.....904111

..114..1691..11472...50995...173606....491533...1215616....2710413....5567530

..200..4059..34350..181336...710976...2269938...6233356...15250675...34054592

..351..9749.102896..644721..2908797..10462235..31868448...85473225..207289059

..616.23422.308419.2294193.11911516..48259083.163014678..479101189.1261310492

.1081.56268.924532.8166441.48807427.222798408.834763824.2688814689.7684922749

#11 by Michel Marcus at Mon Mar 22 03:44:57 EDT 2021

NAME

T(n,k) = is the number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors.

STATUS

proposed

editing

#10 by Jon E. Schoenfield at Sun Mar 21 23:19:21 EDT 2021

STATUS

editing

proposed

#9 by Jon E. Schoenfield at Sun Mar 21 23:19:19 EDT 2021

EXAMPLE

Some solutions for n=4 , k=3:

.. 1.... 2.... 3.... 0.... 0.... 1.... 2.... 3.... 0.... 1.... 2.... 3.... 3.... 1.... 2.... 2

.. 1.... 2.... 1.... 0.... 1.... 0.... 1.... 0.... 3.... 0.... 2.... 2.... 3.... 0.... 3.... 2

.. 2.... 2.... 3.... 0.... 2.... 2.... 3.... 2.... 3.... 0.... 3.... 3.... 3.... 1.... 3.... 0

.. 2.... 0.... 3.... 0.... 3.... 3.... 3.... 3.... 2.... 0.... 3.... 3.... 3.... 1.... 0.... 2

.. 1.... 1.... 2.... 1.... 3.... 3.... 2.... 3.... 0.... 1.... 3.... 3.... 3.... 1.... 2.... 3

.. 0.... 2.... 2.... 1.... 3.... 2.... 1.... 0.... 2.... 1.... 2.... 1.... 1.... 3.... 3.... 3

#8 by Jon E. Schoenfield at Sun Mar 21 23:18:12 EDT 2021

NAME

T(n,k) =Number number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors.

CROSSREFS

Column 1 is A005251(n+5).

Row 1 is A002412(n+1).

AUTHOR

R. H. Hardin , Nov 23 2011

STATUS

proposed

editing

#7 by Robert Israel at Sun Mar 21 19:19:44 EDT 2021

STATUS

editing

proposed

#6 by Robert Israel at Sun Mar 21 19:19:37 EDT 2021

COMMENTS

T(n,k) is the number of lattice points in k*P where P is a polytope of dimension n+2 in R^(n+2) whose vertices are lattice points, and therefore for each n it is an Ehrhart polynomial of degree n+2. This confirms the empirical formulas for the rows. - Robert Israel, Mar 21 2021

STATUS

approved

editing