A026794 -id:A026794 - cp's OEIS frontend

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

1, 0, 1, 2, 5, 7, 16, 20, 40, 55, 97, 124, 235, 287, 482, 654, 1033, 1318, 2137, 2676, 4157, 5439, 7891, 10144, 15280, 19171, 27336, 35652, 49756, 63150, 89342, 111956, 154400, 197413, 264572, 336082, 456724, 568932, 756065, 959566, 1261803, 1576355, 2078267

Offset: 0

Gus Wiseman, Dec 08 2018

nonn

			The a(5) = 16 pairs of integer partitions:
      (51)|(6)
      (42)|(6)
     (411)|(6)
      (33)|(6)
     (321)|(6)
    (3111)|(6)
     (222)|(6)
     (222)|(33)
    (2211)|(6)
    (2211)|(33)
   (21111)|(6)
   (21111)|(33)
  (111111)|(6)
  (111111)|(42)
  (111111)|(33)
  (111111)|(222)

Cf. A265947, A317144, A318915, A322435, A322436, A322439.

g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-i, min(n-i, i))*b(n, i+1), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018

Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n==0, 1, Sum[g[n-i, Min[n-i, i]]*b[n, i+1], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = Sum_{k=1..n-1} A026820(n, k) * A026794(n, k + 1).

A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 7, 2, 0, 1, 12, 1, 0, 0, 1, 19, 4, 2, 0, 0, 1, 30, 3, 1, 0, 0, 0, 1, 45, 8, 1, 2, 0, 0, 0, 1, 67, 7, 4, 1, 0, 0, 0, 0, 1, 97, 15, 3, 1, 2, 0, 0, 0, 0, 1, 139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1, 195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1, 272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jan 19 2013

In other words, T(n,k) is the total number of appearances of k in all partitions of n whose smallest part is k.
The sum of row n equals spt(n), the smallest part partition function (see A092269).
T(n,k) is also the sum of row k in the slice n of tetrahedron A209314.

			Triangle begins:
    1;
    2,  1;
    4,  0, 1;
    7,  2, 0, 1;
   12,  1, 0, 0, 1;
   19,  4, 2, 0, 0, 1;
   30,  3, 1, 0, 0, 0, 1;
   45,  8, 1, 2, 0, 0, 0, 1;
   67,  7, 4, 1, 0, 0, 0, 0, 1;
   97, 15, 3, 1, 2, 0, 0, 0, 0, 1;
  139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1;
  195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1;
  272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  ...
The partitions of 6 with the smallest part in brackets are
..........................
.                      [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]
..........................
There are 19 smallest parts of size 1. Also there are four smallest parts of size 2. Also there are two smallest parts of size 3. There are no smallest part of size 4 or 5. Finally there is only one smallest part of size 6. So row 6 gives 19, 4, 2, 0, 0, 1. The sum of row 6 is 19+4+2+0+0+1 = A092269(6) = 26.

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

Columns 1-3: A000070, A087787, A174455.
Row sums give A092269.
Cf. A026794, A182703, A209314.

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(irem(n, i, 'r')=0, [0$(i-1), r], []); for j from 0
      to n/i do zip((x, y)->x+y, %, [b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 20 2013

b[n_, i_] := b[n, i] = Module[{j, q, r, pc}, If [n == 0 || i<1, 0, {q, r} = QuotientRemainder[n, i]; pc = If[r == 0, Append[Array[0&, i-1], q], {}]; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, b[n-i*j, i-1]}]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

A161364 Triangle read by rows, modified version of A161363; row sums = A000041.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0, 0, 3, 2, 2, 0, 0, 0, 7, 2, 2, 0, 0, 0, 0, 7, 3, 2, 3, 0, 0, 0, 0, 12, 3, 4, 3, 0, 0, 0, 0, 0, 13, 5, 4, 3, 5, 0, 0, 0, 0, 0, 22, 6, 6, 3, 5, 0, 0, 0, 0, 0, 0, 2, 5, 7, 6, 6, 5, 7, 0, 0, 0, 0, 0, 0, 4, 9, 8, 6, 5, 7, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, Jun 07 2009

Row sums = A000041, the partition numbers.

			First few rows of the triangle =
1;
1, 0;
2, 0, 0;
2, 1, 0, 0;
4, 1, 0, 0, 0;
3, 2, 2, 0, 0, 0;
7, 2, 2, 0, 0, 0, 0;
7, 3, 2, 3, 0, 0, 0, 0;
12, 3, 4, 3, 0, 0, 0, 0, 0;
13, 5, 4, 3, 5, 0, 0, 0, 0, 0;
22, 6, 6, 3, 5, 0, 0, 0, 0, 0, 0;
25, 7, 6, 6, 5, 7, 0, 0, 0, 0, 0, 0;
42, 9, 8, 6, 5, 7, 0, 0, 0, 0, 0, 0, 0;
48, 13, 8, 9, 5, 7, 11, 0, 0, 0, 0, 0, 0, 0;
...

A161363, A000041

Given triangle A161363, (the inverse of partition triangle A026794); multiply by (-1), delete right border of 1's, and shift down 1 row inserting a "1" at T(0,0); = triangle M. Let Q = an infinite lower triangular matrix with A000041 as the right border and the rest zeros. Triangle A161364 = M * Q.

A027185 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the least being k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 6, 1, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 1, 12, 2, 1, 0, 0, 0, 0, 0, 1, 14, 4, 1, 0, 0, 0, 0, 0, 0, 1, 22, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 27, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 40, 7, 3, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling

			1,
0, 1,
1, 0, 1,
1, 0, 0, 1,
3, 0, 0, 0, 1,
3, 1, 0, 0, 0, 1,
6, 1, 0, 0, 0, 0, 1,
7, 2, 0, 0, 0, 0, 0, 1,
12, 2, 1, 0, 0, 0, 0, 0, 1,
14, 4, 1, 0, 0, 0, 0, 0, 0, 1,
22, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1,
27, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1,
40, 7, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,

Cf. A027186.

O(n, k) = E(n-k, k)+E(n-k, k+1)+...+E(n-k, n-k), for 2<=2k<=n, E given by A027186.

T(n,k) + A027186(n,k) = A026794(n,k). - R. J. Mathar, Oct 18 2019

A137631 Left border of triangle A137629.

Original entry on oeis.org

1, 2, 4, 7, 11, 18, 26, 39, 55, 79, 106, 150, 196, 267

Offset: 1

Gary W. Adamson, Jan 31 2008

nonn

			a(4) = 7 = (3, 1, 0, 1) dot (1, 1, 2, 3), where (3, 1, 0, 1) = row 4 of the partition triangle A026794 and (1, 1, 2, 3) = the first 4 terms of A000041.

Cf. A000041, A026794, A137629.

A026794 * A000041 = product of the partition triangle and the partitions of n.

A137640 Row sums of triangle A137639.

Original entry on oeis.org

1, 3, 4, 8, 9, 17, 19, 32, 40, 57, 71, 106, 129, 176

Offset: 1

Gary W. Adamson, Jan 31 2008

nonn

			a(4) = 8 = sum of row 4 terms of triangle A137639: (7 + 2 + 0 + 1).
a(4) = 8 = (3, 1, 0, 1) dot (1, 2, 2, 3) = (3 + 2 + 0 + 3); where (3, 1, 0, 1) = row 4 of triangle A026794, the partition triangle and (1, 2, 2, 3) = the first 4 terms of A000005, d(n).

Cf. A000005, A026794, A137639.

A026794 * A000005. (A026794 as an infinite lower triangular matrix and d(n) as a vector).

A161375 Row sums of triangle A161363.

Original entry on oeis.org

1, 0, -1, -2, -4, -5, -9, -11, -17, -21, -32, -38, -58, -70, -97, -122, -170, -206, -284, -349, -463, -573, -752, -919, -1199, -1466, -1869, -2289, -2904, -3518, -4432, -5371, -6676, -8072, -9970, -11985, -14739, -17667, -21522, -25747, -31242, -37176, -44905

Offset: 1

Gary W. Adamson, Jun 07 2009

sign

The partition triangle A026794 * [1, 0, -1, -2, -4, -5,...] = [1, 1, 1, 1,...].

			a(4) = sum of row 4 terms, triangle A161363: (-2, -1, 0, 1).

Cf. A161363, A161364, A000041, A026794

Row sums of triangle A161363.

a(15)-a(43) from Alois P. Heinz, Jun 05 2023

A340928 Least image of A001222 applied to the prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 28 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 4277 are {4,6,15} with images {2,2,2}, so a(4277) = 2.
The prime indices of 8303 are {8,8,9} with images {3,3,2}, so a(8303) = 2.

Positions of 0's are A000079.
Positions of first appearances are A033844.
The version for maximum is A340691.
A003963 multiplies together the prime indices.
A026794 counts partitions by sum and minimum.
A056239 adds up the prime indices.
A061395 selects the greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A001222, A004277, A039900, A062447, A302540, A303975, A324522, A340610.

Table[If[n==1,0,Min@@PrimeOmega/@PrimePi/@First/@FactorInteger[n]],{n,100}]

A112995 Triangular array T(n,k)=number of partitions of n in which the least part is <=k, for 1<=k<=n, n>=1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 9, 10, 10, 10, 11, 11, 13, 14, 14, 14, 14, 15, 15, 19, 20, 21, 21, 21, 21, 22, 22, 26, 28, 29, 29, 29, 29, 29, 30, 30, 37, 39, 40, 41, 41, 41, 41, 41, 42, 42, 50, 53, 54, 55, 55, 55, 55, 55, 55, 56, 56, 68, 72, 74, 75, 76, 76, 76, 76

Offset: 1

Clark Kimberling, Oct 08 2005

The partition sequence A000041 occurs here as T(n,0) and also essentially as T(n,n).

			The first five rows:
1
1 2
2 2 3
3 4 4 5
5 6 6 6 7
Row 5 of A026794, namely 5 1 0 0 1, has partial sums 5 6 6 6 7.

Cf. A000041, A026794.

Row n of T consists of the partial sums of row n of the array in A026794.

cp's OEIS Frontend

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

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A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

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A161364 Triangle read by rows, modified version of A161363; row sums = A000041.

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A027185 Triangular array O by rows: O(n,k) = number of partitions of n into an odd number of parts, the least being k.

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A137631 Left border of triangle A137629.

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A137640 Row sums of triangle A137639.

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A161375 Row sums of triangle A161363.

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A340928 Least image of A001222 applied to the prime indices of n.

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A112995 Triangular array T(n,k)=number of partitions of n in which the least part is <=k, for 1<=k<=n, n>=1.

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