A207379 -id:A207379 - cp's OEIS frontend

A135010 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.

1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 5, 3, 4, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 2, 3, 3, 2, 6, 3, 5, 4, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 17 2007, Mar 21 2008

This is the original sequence of a large number of sequences connected with the section model of partitions.
Here "the n-th section of the set of partitions of any integer greater than or equal to n" (hence "the last section of the set of partitions of n") is defined to be the set formed by all parts that occur as a result of taking all partitions of n and then removing all parts of the partitions of n-1. For integers greater than 1 the structure of a section has two main areas: the head and tail. The head is formed by the partitions of n that do not contain 1 as a part. The tail is formed by A000041(n-1) partitions of 1. The set of partitions of n contains the sets of partitions of the previous numbers. The section model of partitions has several versions according with the ordering of the partitions or with the representation of the sections. In this sequence we use the ordering of A026791.
The section model of partitions can be interpreted as a table of partitions. See also A138121. - Omar E. Pol, Nov 18 2009
It appears that the versions of the model show an overlapping of sections and subsections of the numbers congruent to k mod m into parts >= m. For example:
First generation (the main table):
Table 1.0: Partitions of integers congruent to 0 mod 1 into parts >= 1.
Second generation:
Table 2.0: Partitions of integers congruent to 0 mod 2 into parts >= 2.
Table 2.1: Partitions of integers congruent to 1 mod 2 into parts >= 2.
Third generation:
Table 3.0: Partitions of integers congruent to 0 mod 3 into parts >= 3.
Table 3.1: Partitions of integers congruent to 1 mod 3 into parts >= 3.
Table 3.2: Partitions of integers congruent to 2 mod 3 into parts >= 3.
And so on.
Conjecture:
Let j and n be integers congruent to k mod m such that 0 <= k < m <= j < n. Let h=(n-j)/m. Consider only all partitions of n into parts >= m. Then remove every partition in which the parts of size m appears a number of times < h. Then remove h parts of size m in every partition. The rest are the partitions of j into parts >= m. (Note that in the section model, h is the number of sections or subsections removed), (Omar E. Pol, Dec 05 2010, Dec 06 2010).
Starting from the first row of triangle, it appears that the total numbers of parts of size k in k successive rows give the sequence A000041 (see A182703). - Omar E. Pol, Feb 22 2012
The last section of n contains A187219(n) regions (see A206437). - Omar E. Pol, Nov 04 2012

			Triangle begins:
  [1];
  [1],[2];
  [1],[1],[3];
  [1],[1],[1],[2,2],[4];
  [1],[1],[1],[1],[1],[2,3],[5];
  [1],[1],[1],[1],[1],[1],[1],[2,2,2],[2,4],[3,3],[6];
  ...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in the ordering mentioned in A026791. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j          Diagram          Parts           Parts
---------------------------------------------------------
.                   _
1  1               |_|                1;              1;
.                 _
2  1             | |_               1,              1,
2  2             |_ _|              2;                2;
.               _
3  1           | |                1,              1,
3  2           | |_ _             1,                1,
3  3           |_ _ _|            3;                  3;
.             _
4  1         | |                1,              1,
4  2         | |                1,                1,
4  3         | |_ _ _           1,                  1,
4  4         |   |_ _|          2,2,                2,2,
4  5         |_ _ _ _|          4;                    4;
.           _
5  1       | |                1,              1,
5  2       | |                1,                1,
5  3       | |                1,                  1,
5  4       | |                1,                  1,
5  5       | |_ _ _ _         1,                    1,
5  6       |   |_ _ _|        2,3,                  2,3,
5  7       |_ _ _ _ _|        5;                      5;
.         _
6  1     | |                1,              1,
6  2     | |                1,                1,
6  3     | |                1,                  1,
6  4     | |                1,                  1,
6  5     | |                1,                    1,
6  6     | |                1,                    1,
6  7     | |_ _ _ _ _       1,                      1,
6  8     |   |   |_ _|      2,2,2,                2,2,2,
6  9     |   |_ _ _ _|      2,4,                    2,4,
6  10    |     |_ _ _|      3,3,                    3,3,
6  11    |_ _ _ _ _ _|      6;                        6;
...
(End)

Alois P. Heinz, Rows n = 1..23, flattened
Omar E. Pol, Illustration of the section model of partitions
Omar E. Pol, Illustration of the section model of partitions (2D view)
Omar E. Pol, Illustration of the section model of partitions (3D view)
Robert Price, Mathematica program to generate "Illustration of the section model of partitions"
Robert Price, Mathematica program to generate "Illustration of the section model of partitions (2D view)"

Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A026791, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207383, A207379, A211009.

with(combinat):
T:= proc(m) local b, ll;
      b:= proc(n, i, l)
            if n=0 then ll:=ll, l[]
          else seq(b(n-j, j, [l[], j]), j=i..n)
            fi
          end;
      ll:= NULL; b(m, 2, []); [1$numbpart(m-1)][], ll
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 19 2012

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[ Array[1 &, {PartitionsP[n - 1]}], Sort[ Reverse /@ Select[ IntegerPartitions[n], FreeQ[#, 1] &], less] ] // Flatten; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 14 2013 *)
Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]]~Join~
DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 9}] // Flatten (* Robert Price, May 12 2020 *)

A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 6, 3, 1, 1, 8, 3, 2, 1, 1, 15, 8, 4, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 32, 17, 9, 6, 3, 2, 1, 1, 42, 20, 13, 7, 5, 3, 2, 1, 1, 64, 34, 19, 13, 8, 5, 3, 2, 1, 1, 83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1, 124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Feb 14 2012

Also T(n,k) is the number of parts >= k in the last section of the set of partitions of n. Therefore T(n,1) = A138137(n), the total number of parts in the last section of the set of partitions of n. For calculation of the number of odd/even parts, etc, follow the same rules from A206563.
More generally, let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the entry A206563.
It appears that reversed rows converge to A000041.
It appears that the first differences of row n together with 1 give the row n of triangle A182703 (see example). - Omar E. Pol, Feb 26 2012

			Illustration of initial terms. First six rows of triangle as sums of columns from the last sections of the first six natural numbers (or as sums of columns from the six sections of 6):
.                                         6
.                                         3 3
.                                         4 2
.                                         2 2 2
.                            5              1
.                            3 2              1
.                  4           1              1
.                  2 2           1              1
.          3         1           1              1
.     2      1         1           1              1
.  1    1      1         1           1              1
. --- --- ------- --------- ----------- --------------
A: 1, 2,1, 3,1,1,  6,3,1,1,  8,3,2,1,1,  15,8,4,2,1,1
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|    |/|/|/|/|/|
B: 1, 1,1, 2,0,1,  3,2,0,1,  5,1,1,0,1,   7,4,2,1,0,1
.
A := initial terms of this triangle.
B := initial terms of triangle A182703.
.
Triangle begins:
1;
2,    1;
3,    1,  1;
6,    3,  1,  1;
8,    3,  2,  1,  1;
15,   8,  4,  2,  1,  1;
19,   8,  5,  3,  2,  1,  1;
32,  17,  9,  6,  3,  2,  1,  1;
42,  20, 13,  7,  5,  3,  2,  1,  1;
64,  34, 19, 13,  8,  5,  3,  2,  1,  1;
83,  41, 26, 16, 11,  7,  5,  3,  2,  1,  1;
124, 68, 41, 27, 17, 12,  7,  5,  3,  2,  1,  1;

Columns (1-2): A138137, A138135.
Row sums give A138879.
Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.

From Omar E. Pol, Dec 07 2019: (Start)
From the formula in A138135 (year 2008) we have that:
A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
Also  A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)

More terms from Alois P. Heinz, Feb 17 2012

A058399 Triangle of partial row sums of partition triangle A008284.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1, 11, 10, 7, 4, 2, 1, 15, 14, 11, 7, 4, 2, 1, 22, 21, 17, 12, 7, 4, 2, 1, 30, 29, 25, 18, 12, 7, 4, 2, 1, 42, 41, 36, 28, 19, 12, 7, 4, 2, 1, 56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1, 77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1, 101, 100, 94, 80, 62

Offset: 1

Wolfdieter Lang, Dec 11 2000

T(n,m) is also the number of m-th largest elements in all partitions of n. - Omar E. Pol, Feb 14 2012
It appears that reversed rows converge to A000070. - Omar E. Pol, Mar 10 2012
The row sums give A006128. - Omar E. Pol, Mar 26 2012
T(n,m) is also the number of regions traversed by the m-th column of the section model of partitions with n sections (Cf. A135010, A206437). - Omar E. Pol, Apr 20 2012

			From _Omar E. Pol_, Mar 10 2012: (Start)
Triangle begins:
   1;
   2,  1;
   3,  2,  1;
   5,  4,  2,  1;
   7,  6,  4,  2,  1;
  11, 10,  7,  4,  2,  1;
  15, 14, 11,  7,  4,  2,  1;
  22, 21, 17, 12,  7,  4,  2,  1;
  30, 29, 25, 18, 12,  7,  4,  2,  1;
  42, 41, 36, 28, 19, 12,  7,  4,  2,  1;
  56, 55, 50, 40, 29, 19, 12,  7,  4,  2,  1;
  77, 76, 70, 58, 43, 30, 19, 12,  7,  4,  2,  1;
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-5: A000041(n), A000065(n+1), A004250(n+2), A035300(n-1), A035301(n-1), n >= 1.

Cf. A008284.

b:= proc(n, k) option remember;
      `if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
    end:
T:= (n, m)-> b(n,n) -b(n,m-1):
seq (seq (T(n, m), m=1..n), n=1..15);  # Alois P. Heinz, Apr 20 2012

t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

T(n, m) = Sum_{k=m..n} A008284(n, k).
G.f. for m-th column: Sum_{n>=1} x^(n)/Product_{k=1..n+m-1} (1 - x^k).
T(n, m) = Sum_{k=1..n} A207379(k, m). - Omar E. Pol, Apr 22 2012

A210945 Triangle read by rows: T(n,k) = number of parts in the k-th column of the mirror of the last shell of the partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 1, 7, 1, 11, 3, 1, 15, 3, 1, 22, 6, 3, 1, 30, 7, 4, 1, 42, 11, 7, 3, 1, 56, 13, 9, 4, 1, 77, 20, 15, 8, 3, 1, 101, 23, 18, 10, 4, 1, 135, 33, 27, 17, 8, 3, 1, 176, 40, 34, 22, 11, 4, 1, 231, 54, 47, 33, 18, 8, 3, 1, 297, 65, 58, 42, 24, 11, 4, 1

Offset: 1

Omar E. Pol, Apr 21 2012

For another version see A207379.

			For n = 7 the illustration shows two arrangements of the last shell of the partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 --------
.                  15,3,1
.
We can see that in the right hand picture (the mirror) the number of part for columns 1..3 are 15, 3, 1 therefore row 7 lists 15, 3, 1.
Written as a triangle begins:
1;
2;
3;
5,    1;
7,    1;
11,   3,  1;
15,   3,  1;
22,   6,  3,  1;
30,   7,  4,  1;
42,  11,  7,  3,  1;
56,  13,  9,  4,  1;
77,  20, 15,  8,  3,  1;
101, 23, 18, 10,  4,  1;
135, 33, 27, 17,  8,  3,  1;
176, 40, 34, 22, 11,  4,  1;
231, 54, 47, 33, 18,  8,  3,  1;
297, 65, 58, 42, 24, 11,  4,  1;

Alois P. Heinz, Rows n = 1..70

Column 1 is A000041,n >= 1. Column 2 is A083751. Column 3 is A119907. Row sums give A138137.

Cf. A135010, A138121, A182717, A207379.

More terms from Alois P. Heinz, May 07 2012

A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 3, 2, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 2, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 5, 2, 4, 3, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 2, 2, 3, 3, 2, 6, 2, 5, 3, 4, 4, 8

Offset: 1

Omar E. Pol, Oct 18 2013

The n-th row of triangle lists the parts of the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010.

			Illustration of initial terms (row = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in colexicographic order, see A211992. More generally, in a master model, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j     Diagram          Parts              Parts
---------------------------------------------------------
.         _
1  1     |_|              1;                 1;
.           _
2  1      _| |              1,                 1,
2  2     |_ _|              2;               2;
.             _
3  1         | |              1,                 1,
3  2      _ _| |              1,               1,
3  3     |_ _ _|              3;             3;
.               _
4  1           | |              1,                 1,
4  2           | |              1,               1,
4  3      _ _ _| |              1,             1,
4  4     |_ _|   |            2,2,           2,2,
4  5     |_ _ _ _|              4;           4;
.                 _
5  1             | |              1,                 1,
5  2             | |              1,               1,
5  3             | |              1,             1,
5  4             | |              1,             1,
5  5      _ _ _ _| |              1,           1,
5  6     |_ _ _|   |            3,2,         3,2,
5  7     |_ _ _ _ _|              5;         5;
.                   _
6  1               | |              1,                 1,
6  2               | |              1,               1,
6  3               | |              1,             1,
6  4               | |              1,             1,
6  5               | |              1,           1,
6  6               | |              1,           1,
6  7      _ _ _ _ _| |              1,         1,
6  8     |_ _|   |   |          2,2,2,       2,2,2,
6  9     |_ _ _ _|   |            4,2,       4,2,
6  10    |_ _ _|     |            3,3,       3,3,
6  11    |_ _ _ _ _ _|              6;       6;
...
Triangle begins:
[1];
[1],[2];
[1],[1],[3];
[1],[1],[1],[2,2],[4];
[1],[1],[1],[1],[1],[3,2],[5];
[1],[1],[1],[1],[1],[1],[1],[2,2,2],[4,2],[3,3],[6];
...

Positive terms of A228716.
Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A135010, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207034, A207383, A207379, A211009.

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A135010 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.

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A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

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A058399 Triangle of partial row sums of partition triangle A008284.

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A210945 Triangle read by rows: T(n,k) = number of parts in the k-th column of the mirror of the last shell of the partitions of n.

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A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

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