A196931 -id:A196931 - cp's OEIS frontend

A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of every integer is reversed with respect to A026792. For example: in A026792 the partitions of 3 are listed as [3], [2, 1], [1, 1, 1], however here the partitions of 3 are listed as [1, 1, 1], [2, 1], [3].
Row n has length A006128(n). Row sums give A066186. Right border gives A000027. The equivalent sequence for compositions (ordered partitions) is A228525. - Omar E. Pol, Aug 24 2013
The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic. The equivalent sequence for partitions as (weakly) increasing lists and lexicographic order is A026791. - Joerg Arndt, Sep 02 2013

			From _Omar E. Pol_, Aug 24 2013: (Start)
Illustration of initial terms:
-----------------------------------------
n      Diagram          Partition
-----------------------------------------
.       _
1      |_|              1;
.       _ _
2      |_| |            1, 1,
2      |_ _|            2;
.       _ _ _
3      |_| | |          1, 1, 1,
3      |_ _| |          2, 1,
3      |_ _ _|          3;
.       _ _ _ _
4      |_| | | |        1, 1, 1, 1,
4      |_ _| | |        2, 1, 1,
4      |_ _ _| |        3, 1,
4      |_ _|   |        2, 2,
4      |_ _ _ _|        4;
.       _ _ _ _ _
5      |_| | | | |      1, 1, 1, 1, 1,
5      |_ _| | | |      2, 1, 1, 1,
5      |_ _ _| | |      3, 1, 1,
5      |_ _|   | |      2, 2, 1,
5      |_ _ _ _| |      4, 1,
5      |_ _ _|   |      3, 2,
5      |_ _ _ _ _|      5;
.       _ _ _ _ _ _
6      |_| | | | | |    1, 1, 1, 1, 1, 1,
6      |_ _| | | | |    2, 1, 1, 1, 1,
6      |_ _ _| | | |    3, 1, 1, 1,
6      |_ _|   | | |    2, 2, 1, 1,
6      |_ _ _ _| | |    4, 1, 1,
6      |_ _ _|   | |    3, 2, 1,
6      |_ _ _ _ _| |    5, 1,
6      |_ _|   |   |    2, 2, 2,
6      |_ _ _ _|   |    4, 2,
6      |_ _ _|     |    3, 3,
6      |_ _ _ _ _ _|    6;
...
Triangle begins:
[1];
[1,1], [2];
[1,1,1], [2,1], [3];
[1,1,1,1], [2,1,1], [3,1], [2,2], [4];
[1,1,1,1,1], [2,1,1,1], [3,1,1], [2,2,1], [4,1], [3,2], [5];
[1,1,1,1,1,1], [2,1,1,1,1], [3,1,1,1], [2,2,1,1], [4,1,1], [3,2,1], [5,1], [2,2,2], [4,2], [3,3], [6];
(End)
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A334437) begins:
    1
    2
    4   3
    8   6   5
   16  12  10   9   7
   32  24  20  18  14  15  11
   64  48  40  36  28  30  22  27  21  25  13
  128  96  80  72  56  60  44  54  42  50  26  45  33  35  17
(End)

Joerg Arndt, Table of n, a(n) for n = 1..10000
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A026791, A141285, A194446, A228531.
The graded reversed version is A026792.
The length-sensitive refinement is A036037.
The version for reversed partitions is A080576.
Partition lengths are A193173.
Partition maxima are A194546.
Partition minima are A196931.
The version for compositions is A228525.
The Heinz numbers of these partitions are A334437.
Cf. A036036, A080577, A193073, A228100, A296150, A331581, A334301, A334302, A334436, A334439, A334442.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Join@@Table[Sort[IntegerPartitions[n],colex],{n,0,6}] (* Gus Wiseman, May 10 2020 *)

gen_part(n)=
{  /* Generate partitions of n as weakly increasing lists (order is lex): */
    my(ct = 0);
    my(m, pt);
    my(x, y);
    \\ init:
    my( a = vector( n + (n<=1) ) );
    a[1] = 0;  a[2] = n;  m = 2;
    while ( m!=1,
        y = a[m] - 1;
        m -= 1;
        x = a[m] + 1;
        while ( x<=y,
            a[m] = x;
            y = y - x;
            m += 1;
        );
        a[m] = x + y;
        pt = vector(m, j, a[j]);
    /* for A026791 print partition: */
\\        for (j=1, m, print1(pt[j],", ") );
    /* for A211992 print partition as weakly decreasing list (order is colex): */
        forstep (j=m, 1, -1, print1(pt[j],", ") );
        ct += 1;
    );
    return(ct);
}
for(n=1, 10, gen_part(n) );
\\ Joerg Arndt, Sep 02 2013

A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 4, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 4, 1, 2, 3, 1, 5, 2, 2, 2, 2, 4, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 5

Offset: 1

Clark Kimberling

Differs from A080576 in a(18): Here, (...,1+3,2+2,4), there (...,2+2,1+3,4).
The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is lexicographic (see example). - Joerg Arndt, Sep 03 2013
The equivalent sequence for compositions (ordered partitions) is A228369. - Omar E. Pol, Oct 19 2019

			First six rows are:
[[1]];
[[1, 1], [2]];
[[1, 1, 1], [1, 2], [3]];
[[1, 1, 1, 1], [1, 1, 2], [1, 3], [2, 2], [4]];
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 4], [2, 3], [5]];
[[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 2, 2], [1, 1, 4], [1, 2, 3], [1, 5], [2, 2, 2], [2, 4], [3, 3], [6]];
...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
----------------------------------
.                     Ordered
n  j      Diagram     partition j
----------------------------------
.               _
1  1           |_|    1;
.             _ _
2  1         | |_|    1, 1,
2  2         |_ _|    2;
.           _ _ _
3  1       | | |_|    1, 1, 1,
3  2       | |_ _|    1, 2,
3  3       |_ _ _|    3;
.         _ _ _ _
4  1     | | | |_|    1, 1, 1, 1,
4  2     | | |_ _|    1, 1, 2,
4  3     | |_ _ _|    1, 3,
4  4     |   |_ _|    2, 2,
4  5     |_ _ _ _|    4;
...
(End)

Alois P. Heinz, Rows n = 1..19, flattened
Wikiversity, Lexicographic and colexicographic order

Row lengths are given in A006128.
Partition lengths are in A193173.
Other partition orderings: A026792, A036037, A080577, A125106, A139100, A181087, A181317, A182937, A228100, A240837, A242628.
Row lengths are A000041.
Partition sums are A036042.
Partition minima are A196931.
Partition maxima are A194546.
The reflected version is A211992.
The length-sensitive version (sum/length/lex) is A036036.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A193073.
Compositions under the same ordering (sum/lex) are A228369.
The reverse-lexicographic version (sum/revlex) is A228531.
The Heinz numbers of these partitions are A334437.
Cf. A049085, A103921, A112798, A115623, A129129, A331581, A334435, A334439, A334442.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, l[]
          else seq(b(n-i, [l[], i]), i=`if`(l=[],1,l[-1])..n)
            fi
          end;
      ll:= NULL; b(n, []); ll
    end:
seq(T(n), n=1..8);  # Alois P. Heinz, Jul 16 2011

T[n0_] := Module[{b, ll}, b[n_, l_] := If[n == 0, ll = Join[ll, l], Table[ b[n - i, Append[l, i]], {i, If[l == {}, 1, l[[-1]]], n}]]; ll = {}; b[n0, {}]; ll]; Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Aug 05 2015, after Alois P. Heinz *)
Table[DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions[n]], x_ /; x == 0, 2], {n, 7}] // Flatten (* Robert Price, May 18 2020 *)

t = [[[]]]
for n in range(1, 10):
    p = []
    for minp in range(1, n):
        p += [[minp] + pp for pp in t[n-minp] if min(pp) >= minp]
    t.append(p + [[n]])
print(t)
# Andrey Zabolotskiy, Oct 18 2019

A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Dec 01 2010

Triangle read by rows in which row n lists the smallest parts of all partitions of n in the order produced by the shell model of partitions of A138121.
Also, row n lists the "filler parts" of all partition of n. For more information see A182699.
Row n has length A000041(n). Row sums give A046746. Column 1 gives A001477. The last A000041(n-1) terms of row n are ones, n >= 1.

			For row 10, see the illustration of the link.
Triangle begins:
  0,
  1,
  2,1,
  3,1,1,
  4,2,1,1,1,
  5,2,1,1,1,1,1,
  6,3,2,2,1,1,1,1,1,1,1,
  7,3,2,2,1,1,1,1,1,1,1,1,1,1,1,
  8,4,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  9,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  ...

Omar E. Pol, Illustration of the smallest parts of all partitions of 10 (colored blue) in the shell model of partitions

Cf. A026807, A135010, A138121, A141285, A182699, A182709.

Mirror of triangle A196931.

Name simplified and more terms from Omar E. Pol, Oct 21 2011

A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 4, 1, 2, 3, 2, 4, 3, 5, 1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 3, 5, 4, 7, 3, 6, 5, 9

Offset: 1

Omar E. Pol, Dec 10 2011

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

			For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Wikiversity, Lexicographic and colexicographic order

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the n-th integer partition in colexicographic order (A211992):
- The maximum of y is a(n).
- The length of y is A193173(n).
- The minimum of y is A196931(n).
- The Heinz number of y is A334437(n).
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* Gus Wiseman, May 31 2020 *)

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Definition corrected by Omar E. Pol, Sep 12 2013

A196025 Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

Original entry on oeis.org

0, 0, 0, 2, 5, 16, 30, 63, 108, 189, 298, 483, 720, 1092, 1582, 2297, 3225, 4551, 6244, 8592, 11590, 15622, 20741, 27536, 36066, 47198, 61150, 79077, 101391, 129808, 164934, 209213, 263745, 331807, 415229, 518656, 644719, 799926, 988432, 1218979

Offset: 1

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182709. Total sum of emergent parts in all partitions of all numbers <= n.

Also total sum of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196039, A196930, A196931, A198381.

a(n) = A066186(n) - A196039(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)). - Vaclav Kotesovec, Jul 06 2019

A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 2, 3, 6, 2, 2, 3, 7, 2, 2, 2, 2, 3, 4, 8, 2, 2, 2, 2, 3, 3, 4, 9, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 10, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6, 13

Offset: 1

Omar E. Pol, Oct 21 2011

For n >= 2, row n lists the parts of the head of the last section of the set of partitions of n, except the emergent parts.

Also 1 together with the integers > 1 of A196931.

			Written as a triangle:
1,
2,
3,
2,4,
2,5,
2,2,3,6
2,2,3,7,
2,2,2,2,3,4,8,
2,2,2,2,3,3,4,9,
2,2,2,2,2,2,2,3,3,4,5,10,
2,2,2,2,2,2,2,2,3,3,3,4,5,11,
2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12,
2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13,
...
Row n has length A002865(n), n >= 2. The sum of row n is A182708(n), n >= 2. The number of 2's in row n is A002865(n-2), n >= 4. Right border of triangle gives A000027.

Alois P. Heinz, Rows n = 1..33, flattened

Where records occur give A000041.

Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931.

p:= (f, g)-> zip((x, y)->x+y, f, g, 0):
b:= proc(n, i) option remember; local g, j, r;
      if n=0 then [1] elif i<2 then [0]
    else r:= b(n, i-1);
         for j to n/i do g:= b(n-i*j, i-1);
           r:= p(p(r, [0$i, g[1]]), subsop(1=0, g));
         od; r
      fi
    end:
T:= proc(n) local l; l:= b(n$2);
      `if`(n=1, 1, seq(i$l[i+1], i=2..nops(l)-1))
    end:
seq(T(n), n=1..16);  # Alois P. Heinz, May 30 2013

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{ g, j, r}, Which[n == 0, {1}, i<2, {0}, True, r = b[n, i-1]; For[j = 1, j <= n/i, j++, g = b[n-i*j, i-1]; r = p[p[r, Append[Array[0&, i], g // First]], ReplacePart[g, 1 -> 0]]]; r]]; T[n_] := Module[{l}, l = b[n, n]; If[n == 1, {1}, Table[Array[i&, l[[i+1]]], {i, 2, Length[l]-1}] // Flatten]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 54, 81, 128, 184, 273, 385, 549, 754, 1048, 1412, 1917, 2547, 3392, 4444, 5837, 7556, 9791, 12553, 16086, 20429, 25935, 32665, 41108, 51404, 64190, 79721, 98882, 122043, 150417, 184618, 226239

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182699. Total number of emergent parts in all partitions of the numbers <= n.

Also total number of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Cf. A000041, A000065, A000070, A006128, A026905, A093694, A096541, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196930, A196931.

a(n) = A096541(n) - A000065(n) = 1 + A096541(n) - A000041(n) = 1 + A006128(n) - A000070(n).

a(n) = A006128(n) - A026905(n), n >= 1.

A196039 Total sum of the smallest part of every partition of every shell of n.

Original entry on oeis.org

0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Partial sums of A046746.

Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012

			For n = 5 the seven partitions of 5 are:
5
3         + 2
4             + 1
2     + 2     + 1
3         + 1 + 1
2     + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
.
The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
.
The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
.
thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.

b:= proc(n, i) option remember;
     `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(nAlois P. Heinz, Apr 03 2012

b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

a(n) = A066186(n) - A196025(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)). - Vaclav Kotesovec, Jul 06 2019

A244966 Triangle read by rows: T(n,k) is the difference between the largest and the smallest part of the k-th partition in the list of colexicographically ordered partitions of n, with n>=1 and 1<=k<=p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 3, 1, 0, 0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0, 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0, 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0, 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 1, 3, 2, 5, 4, 3, 7, 1, 3, 2, 5, 0, 3, 1, 0

Offset: 1

Omar E. Pol, Jul 18 2014

The number of t's in row n gives A097364(n,t), with n>=1 and 0<=t

Rows converge to A244967, which is A141285 - 1.

Row n has length A000041(n).

Row sums give A116686.

			Triangle begins:
0;
0, 0;
0, 1, 0;
0, 1, 2, 0, 0;
0, 1, 2, 1, 3, 1, 0;
0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0;
0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0;
0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0;
...
For n = 6 we have:
--------------------------------------------------------
.                        Largest  Smallest   Difference
k    Partition of 6        part     part       T(6,k)
--------------------------------------------------------
1:  [1, 1, 1, 1, 1, 1]      1    -    1     =     0
2:  [2, 1, 1, 1, 1]         2    -    1     =     1
3:  [3, 1, 1, 1]            3    -    1     =     2
4:  [2, 2, 1, 1]            2    -    1     =     1
5:  [4, 1, 1]               4    -    1     =     3
6:  [3, 2, 1]               3    -    1     =     2
7:  [5, 1]                  5    -    1     =     4
8:  [2, 2, 2]               2    -    2     =     0
9:  [4, 2]                  4    -    2     =     2
10: [3, 3]                  3    -    3     =     0
11: [6]                     6    -    6     =     0
--------------------------------------------------------
So the 6th row of triangle is [0,1,2,1,3,2,4,0,2,0,0] and the row sum is A116686(6) = 15.
Note that in the 6th row there are four 0's so A097364(6,0) = 4, there are two 1's so A097364(6,1) = 2, there are three 2's so A097364(6,2) = 3, there is only one 3 so A097364(6,3) = 1, there is only one 4 so A097364(6,4) = 1 and there are no 5's so A097364(6,5) = 0.

G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Cf. A000005, A000041, A008805, A049820, A097364, A116686, A128508, A135010, A141285, A196931, A218567-A218573, A244967.

T(n,k) = A141285(k) - A196931(n,k), n>=1, 1<=k<=A000041(n).

Showing 1-9 of 9 results.

cp's OEIS Frontend

A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

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A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

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A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

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A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

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A196025 Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

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A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

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A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

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A196039 Total sum of the smallest part of every partition of every shell of n.

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