OFFSET
1,4
COMMENTS
For the definition of "section" of the set of partitions of n see A135010.
Also, column 1 gives the number of partitions of n-1. For k >= 2, row n lists the number of k's in all partitions of n that do not contain 1 as a part.
From Omar E. Pol, Feb 12 2012: (Start)
It appears that reversed rows converge to A002865.
It appears that row n is also the base of an isosceles triangle in which the column sums give the partition numbers A000041 in descending order starting with p(n-1) = A000041(n-1). Example for n = 7:
.
. 1,
. 1, 0, 1,
. 4, 2, 1, 0, 1,
11, 3, 2, 1, 1, 0, 1,
---------------------
11, 7, 5, 3, 2, 1, 1,
.
It appears that in row n starts an infinite trapezoid in which column sums always give the number of partitions of n-1. Example for n = 7:
.
11, 3, 2, 1, 1, 0, 1,
. 8, 3, 3, 1, 1, 0, 1,
. 6, 2, 2, 1, 1, 0, 1,
. 5, 3, 2, 1, 1, 0, 1,
. 4, 2, 2, 1, 1, 0, 1,
. 5, 2, 2, 1, 1, 0,...
. 4, 2, 2, 1, 1,...
. 4, 2, 2, 1,...
. 4, 2, 2,...
. 4, 2,...
. 4,...
.
The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.
It appears that the first term of row n is one of the vertices of an infinite isosceles triangle in which column sums give the partition numbers A000041 in ascending order starting with p(n-1) = A000041(n-1). Example for n = 7:
11,
. 8,
. 7, 6,
. 6, 5,
. 10, 5, ...
. 10, ...
. 10, ...
-------------------
11, 15, 22, 30, ...
(End)
It appears that row n lists the first differences of the row n of triangle A207031 together with 1 (as the final term of row n). - Omar E. Pol, Feb 26 2012
More generally T(n,k) is the number of occurrences of k in the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Oct 21 2013
LINKS
EXAMPLE
Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
. _ _ _ _ _ _ _
. (7) (7) |_ _ _ _ |
. (4+3) (4+3) |_ _ _ _|_ |
. (5+2) (5+2) |_ _ _ | |
. (3+2+2) (3+2+2) |_ _ _|_ _|_ |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) |_|
. ----------------
. 19,8,5,3,2,1,1 --> Row 7 of triangle A207031.
. |/|/|/|/|/|/|
. 11,3,2,1,1,0,1 --> Row 7 of this triangle.
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So, for k = 1..7, row 7 gives: 11, 3, 2, 1, 1, 0, 1.
Triangle begins:
1;
1, 1;
2, 0, 1;
3, 2, 0, 1;
5, 1, 1, 0, 1;
7, 4, 2, 1, 0, 1;
11, 3, 2, 1, 1, 0, 1;
15, 8, 3, 3, 1, 1, 0, 1;
22, 7, 6, 2, 2, 1, 1, 0, 1;
30, 15, 6, 5, 3, 2, 1, 1, 0, 1;
42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1;
56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
...
MAPLE
p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local g;
if n=0 then [1]
elif n<2 or i<2 then [0]
else g:= `if`(i>n, [0], b(n-i, i));
p(p([0$j=2..i, g[1]], b(n, i-1)), g)
fi
end:
h:= proc(n) option remember;
`if`(n=0, 1, b(n, n)[1]+h(n-1))
end:
T:= proc(n) h(n-1), b(n, n)[2..n][] end:
seq(T(n), n=1..20); # Alois P. Heinz, Feb 19 2012
MATHEMATICA
p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1}, n<2 || i<2, {0}, True, g = If [i>n, {0}, b[n-i, i]]; p[p[Append[Array[0&, i-1], g[[1]]], b[n, i-1]], g]]]; h[n_] := h[n] = If[n == 0, 1, b[n, n][[1]] + h[n-1]]; t[n_] := {h[n-1], Sequence @@ b[n, n][[2 ;; n]]}; Table[t[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)
Table[{PartitionsP[n-1]}~Join~Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], k], {k, 2, n}], {n, 1, 12}] // Flatten (* Robert Price, May 15 2020 *)
CROSSREFS
AUTHOR
Omar E. Pol, Nov 28 2010
STATUS
approved