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%I #108 May 16 2020 01:13:01
%S 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,
%T 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,
%U 0,1,56,27,14,10,5,5,2,2,1,1,0,1
%N Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.
%C For the definition of "section" of the set of partitions of n see A135010.
%C 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.
%C From _Omar E. Pol_, Feb 12 2012: (Start)
%C It appears that reversed rows converge to A002865.
%C 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:
%C .
%C . 1,
%C . 1, 0, 1,
%C . 4, 2, 1, 0, 1,
%C 11, 3, 2, 1, 1, 0, 1,
%C ---------------------
%C 11, 7, 5, 3, 2, 1, 1,
%C .
%C 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:
%C .
%C 11, 3, 2, 1, 1, 0, 1,
%C . 8, 3, 3, 1, 1, 0, 1,
%C . 6, 2, 2, 1, 1, 0, 1,
%C . 5, 3, 2, 1, 1, 0, 1,
%C . 4, 2, 2, 1, 1, 0, 1,
%C . 5, 2, 2, 1, 1, 0,...
%C . 4, 2, 2, 1, 1,...
%C . 4, 2, 2, 1,...
%C . 4, 2, 2,...
%C . 4, 2,...
%C . 4,...
%C .
%C The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.
%C 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:
%C 11,
%C . 8,
%C . 7, 6,
%C . 6, 5,
%C . 10, 5, ...
%C . 10, ...
%C . 10, ...
%C -------------------
%C 11, 15, 22, 30, ...
%C (End)
%C 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
%C 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
%H Alois P. Heinz, <a href="/A182703/b182703.txt">Rows n = 1..141, flattened</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the section model of partitions, Figure 1 (n = 1..6)</a>, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Figure 2 (2D view, n = 1..10)</a>, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Figure 3 (3D view, n = 1..9)</a>
%F It appears that T(n,k) = A207032(n,k) - A207032(n,k+2). - Omar E. Pol, Feb 26 2012
%e 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:
%e . _ _ _ _ _ _ _
%e . (7) (7) |_ _ _ _ |
%e . (4+3) (4+3) |_ _ _ _|_ |
%e . (5+2) (5+2) |_ _ _ | |
%e . (3+2+2) (3+2+2) |_ _ _|_ _|_ |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) | |
%e . (1) (1) |_|
%e . ----------------
%e . 19,8,5,3,2,1,1 --> Row 7 of triangle A207031.
%e . |/|/|/|/|/|/|
%e . 11,3,2,1,1,0,1 --> Row 7 of this triangle.
%e .
%e 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.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 0, 1;
%e 3, 2, 0, 1;
%e 5, 1, 1, 0, 1;
%e 7, 4, 2, 1, 0, 1;
%e 11, 3, 2, 1, 1, 0, 1;
%e 15, 8, 3, 3, 1, 1, 0, 1;
%e 22, 7, 6, 2, 2, 1, 1, 0, 1;
%e 30, 15, 6, 5, 3, 2, 1, 1, 0, 1;
%e 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1;
%e 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
%e ...
%p p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
%p b:= proc(n,i) option remember; local g;
%p if n=0 then [1]
%p elif n<2 or i<2 then [0]
%p else g:= `if`(i>n, [0], b(n-i, i));
%p p(p([0$j=2..i, g[1]], b(n, i-1)), g)
%p fi
%p end:
%p h:= proc(n) option remember;
%p `if`(n=0, 1, b(n, n)[1]+h(n-1))
%p end:
%p T:= proc(n) h(n-1), b(n, n)[2..n][] end:
%p seq(T(n), n=1..20); # _Alois P. Heinz_, Feb 19 2012
%t 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 *)
%t 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 *)
%Y Row sums give A138137. Where records occur is A134869.
%Y Columns 1-10: A000041, A182712-A182714, A206555-A206560.
%Y Sub-triangles (1-11): A023531, A129186, A194702-A194710
%Y Cf. A066633, A135010, A182742, A182743, A194812, A206563, A207031, A207032, A206437, A211025.
%K nonn,tabl,look
%O 1,4
%A _Omar E. Pol_, Nov 28 2010
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