This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A238883 Array:  row n gives number of times each upper triangular partition U(p) occurs as p ranges through the partitions of n. 4
 1, 2, 3, 4, 1, 4, 3, 8, 1, 2, 10, 3, 2, 14, 5, 2, 1, 20, 3, 4, 2, 1, 30, 3, 2, 1, 6, 36, 13, 2, 3, 2, 52, 10, 4, 6, 3, 2, 70, 9, 9, 4, 6, 3, 94, 16, 6, 5, 10, 2, 2, 122, 24, 4, 8, 1, 12, 2, 2, 1, 160, 33, 4, 12, 6, 4, 9, 2, 1, 206, 37, 18, 14, 6, 2, 6, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Suppose that p is a partition.  Let u, v, w be the number of 1's above, on, and below the principal antidiagonal, respectively, of the Ferrers matrix of p defined at A237981.  The upper triangular partition of p, denoted by U(p), is {u,v} if w = 0 and {u,v,w} otherwise.  In row n, the counted partitions are taken in Mathematica order (i.e., reverse lexicographic).  A000041 = sum of numbers in row n, and A238884(n) = (number of numbers in row n) = number of upper triangular partitions of n. LINKS Clark Kimberling, Table of n, a(n) for n = 1..300 Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers EXAMPLE First 12 rows: 1 2 3 4 .. 1 4 .. 3 8 .. 1 .. 2 10 . 3 .. 2 14 . 5 .. 2 .. 1 20 . 3 .. 4 .. 2 .. 1 30 . 3 .. 2 .. 1 .. 6 36 . 13 . 2 .. 3 .. 2 52 . 10 . 4 .. 6 .. 3 .. 2 Row 6 arises as follows:  there are 3 upper triangular (UT) partitions:  51, 33, 321, of which 51 is produced from the 8 partitions  6, 51, 42, 411, 3111, 2211, 21111, and 111111, while the UT partition 33 is produced from the single partition 321, and the only other UT partition of 6, namely 321, is produced from the partitions 33 and 222.  (For example, the rows of the Ferrers matrix of 222 are (1,1,0), (1,1,0), (1,1,0), with principal antidiagonal (0,1,1), so that u = 3, v = 2, w = 1.) MATHEMATICA ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; ut[list_] := Select[Map[Total[Flatten[#]] &, {LowerTriangularize[#, -1], Diagonal[#], UpperTriangularize[#, 1]}] &[Reverse[ferrersMatrix[list]]], # > 0 &]; t[n_] := #[[Reverse[Ordering[PadRight[Map[First[#] &, #]]]]]] &[  Tally[Map[Reverse[Sort[#]] &, Map[ut, IntegerPartitions[n]]]]] u[n_] := Table[t[n][[k]][[1]], {k, 1, Length[t[n]]}]; v[n_] := Table[t[n][[k]][[2]], {k, 1, Length[t[n]]}]; TableForm[Table[t[n], {n, 1, 12}]] z = 20; Table[Flatten[u[n]], {n, 1, z}] Flatten[Table[u[n], {n, 1, z}]] Table[v[n], {n, 1, z}] Flatten[Table[v[n], {n, 1, z}]] (* A238883 *) Table[Length[v[n]], {n, 1, z}]  (* A238884 *) (* Peter J. C. Moses, Mar 04 2014 *) CROSSREFS Cf. A238884, A238885, A238886, A237981, A000041. Sequence in context: A318308 A003324 A110630 * A257053 A129717 A317088 Adjacent sequences:  A238880 A238881 A238882 * A238884 A238885 A238886 KEYWORD nonn,tabf,easy AUTHOR Clark Kimberling, Mar 06 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.

Last modified November 17 22:58 EST 2018. Contains 317279 sequences. (Running on oeis4.)