login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A237980 Array:  row n gives the number of distinct square partitions of n; see Comments. 2
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 25, 28, 32, 36, 44, 49, 60, 66, 80, 89, 103, 115, 132, 147, 168, 188, 212, 236, 269, 301, 344, 385, 437, 485, 549, 606, 678, 751, 837, 926, 1031, 1133, 1263, 1389, 1541, 1696, 1889, 2068, 2306, 2529 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Suppose that p is a partition of n.  Let m X m be the size of its Ferrers matrix, f(p), defined at A237981.  Then f(p) consists of ceiling(m/2) concentric squares, where the innermost square is a single point if m is odd.  The square partition of p is introduced here as the partition [x(1), x(2) ,..., x(k)], where x(i) is the number of 1s in the i-th concentric square, where the squares are taken in order starting with the outermost.

LINKS

Table of n, a(n) for n=1..58.

EXAMPLE

The 7 square partitions of 12 are as follows:  [12], [11,1], [10,2], [9,3], [8,3,1], [8,4], [7,4,1].  The Ferrers matrix of the partition [4,3,3,1,1] of 12 is shown here:

1 . 1 . 1 . 1 . 0

1 . 1 . 1 . 0 . 0

1 . 1 . 1 . 0 . 0

1 . 0 . 0 . 0 . 0

1 . 0 . 0 . 0 . 0.

The outermost square has 8 1s, the next has 3 1s, and the innermost, 1 1, so that [8,3,1] is a square partition of 12.

MATHEMATICA

z=20;

ferrersMatrix[list_]:=PadRight[Map[Table[1, {#}]&, #], {#, #}&[Max[#, Length[#]]]]&[list];

sqPart[list_]:=DeleteCases[Total[{Total[LowerTriangularize[#]+ Transpose[UpperTriangularize[#, 1]]]&[Reverse[LowerTriangularize[#]]], Reverse[Total[Transpose[ LowerTriangularize[#]]+UpperTriangularize[#, 1]]]&[Reverse[UpperTriangularize[#, 1]]]}&[ferrersMatrix[list]]], 0];

sqParts[n_]:=#[[Reverse[Ordering[PadRight[#]]]]]&[DeleteDuplicates[Map[sqPart, IntegerPartitions[n]]]]

Flatten[sq=Map[sqParts[#]&, Range[z]]] (*A237985*)

Map[Length, sq] (*A237980*)

(* Peter J. C. Moses, Feb 19 2014 *)

CROSSREFS

Cf. A237985.

Sequence in context: A025148 A096749 A036821 * A026798 A185325 A125890

Adjacent sequences:  A237977 A237978 A237979 * A237981 A237982 A237983

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling and Peter J. C. Moses, Feb 25 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 18 16:11 EDT 2017. Contains 290727 sequences.