A115622 -id:A115622 - cp's OEIS frontend

A115623 Irregular triangle read by rows: row n lists numbers of distinct parts of partitions of n in Mathematica order.

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

Offset: 0

Franklin T. Adams-Watters, Jan 25 2006

The row length sequence of this table is p(n)=A000041(n) (number of partitions).
In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [3,1,1,1] as {1,3} and count the elements, here 2.
Rows are the same as the rows of A103921, but in reverse order.

			Triangle starts:
0
1
1, 1
1, 2, 1
1, 2, 1, 2, 1
1, 2, 2, 2, 2, 2, 1
1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 1
1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1
1, 2, 2, 2, 2, 3, 2, 1, 3, 2, 3, 2, 2, 2, 3, 3, 2, 1, 2, 2, 2, 1
1, 2, 2, 2, 2, 3, 2, 2, ...
a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., [3,1^2], which has two distinct parts, namely 1 and 3.

Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).

Cf. A080577, A000041, A103921, A115622, row sums A000070.

Table[Length /@ Union /@ IntegerPartitions[n], {n, 0, 8}] // Flatten  (* Robert Price, Jun 11 2020 *)

a(n, m) = number of distinct parts of the m-th partition of n in Mathematica order; n >= 0, m = 1..p(n) = A000041(n).

Edited and corrected by Franklin T. Adams-Watters, May 29 2006

A115621 Signature of partitions in Abramowitz and Stegun order.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Jan 25 2006

The signature of a multiset is a partition consisting of the repetition factors of the original partition. Regarding a partition as a multiset, the signature of a partition is defined. E.g., [1,1,3,4,4] = [1^2,3^1,4^2], so the repetition factors are 2,1,2, making the signature [1,2,2] = [1,2^2]. Partitions are written here in increasing part size, so [1,2^2] is 1,2,2, not 2,2,1. - Edited by Franklin T. Adams-Watters, Jul 09 2012

The sum (or order) of the signature is the number of parts of the original partition and the number of parts of the signature is the number of distinct parts of the original partition.

			[1];
[1], [2];
[1], [1,1], [3];
[1], [1,1], [2], [1,2], [4];
...
From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
Extending the triangle to rows 5 and 6 where row headings indicate the number of elements in the underlying partitions. Brackets group the multiplicities of a single partition.
    row 5         row 6
1:  [1]           [1]
2:  [1,1] [1,1]   [1,1] [1,1] [2]
3:  [1,2] [1,2]   [1,2] [1,1,1] [3]
4:  [1,3]         [1,3] [2,2]
5:  [5]           [1,4]
6:                [6]
(End)

Robert Price, Table of n, a(n) for n = 1..8266 (first 20 rows).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A036036, A113787, A115622, A103921 (part counts), A000070 (row counts).

(* row[] and triangle[] compute structured rows of the triangle as laid out above *)
mL[pL_] := Map[Last[Transpose[Tally[#]]]&, pL]
row[n_] := Map[Map[Sort, mL[#]]&, GatherBy[Map[Sort, IntegerPartitions[n]], Length]]
triangle[n_] := Map[row, Range[n]]
a115621[n_]:= Flatten[triangle[n]]
Take[a115621[8],105] (* data *)  (* Hartmut F. W. Hoft, Apr 25 2015 *)
Map[Sort[#, Less] &, Table[Last /@ Transpose /@ Tally /@ Sort[Reverse /@ IntegerPartitions[n]], {n, 8}], 2]

from collections import Counter
def A115621_row(n):
    h = lambda p: sorted(Counter(p).values())
    return flatten([h(p) for k in (0..n) for p in Partitions(n, length=k)])
for n in (1..10): print(A115621_row(n)) # Peter Luschny, Nov 02 2019

A115262 Correlation triangle for n+1.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 14, 11, 5, 6, 14, 20, 20, 14, 6, 7, 17, 26, 30, 26, 17, 7, 8, 20, 32, 40, 40, 32, 20, 8, 9, 23, 38, 50, 55, 50, 38, 23, 9, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 11, 29, 50, 70, 85, 91, 85, 70, 50, 29, 11

Offset: 0

Paul Barry, Jan 18 2006

This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - Philippe Deléham, Nov 26 2009
From Clark Kimberling, Feb 07 2011: (Start)
Also the accumulation array of min{n,k}, when formatted as a rectangle.
This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
The accumulation array of A115262 is A185957. (End)
From Clark Kimberling, Dec 22 2011: (Start)
As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...).  See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)

			Triangle begins
  1;
  2,  2;
  3,  5,  3;
  4,  8,  8,  4;
  5, 11, 14, 11,  5;
  6, 14, 20, 20, 14,  6;
  ...
When formatted as a square matrix:
  1,  2,  3,  4,  5, ...
  2,  5,  8, 11, 14, ...
  3,  8, 14, 20, 26, ...
  4, 11, 20, 30, 40, ...
  5, 14, 26, 40, 55, ...
  ...

Cf. A000027, A202673, A271916.
For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).
Diagonals (1,5,...): A000330 (square pyramidal numbers),
diagonals (2,8,...): A007290,
diagonals (3,11,...): A051925,
diagonals (4,14,...): A159920,
antidiagonal sums: A001752.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
(* Clark Kimberling, Dec 22 2011 *)

Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - N. J. A. Sloane, Apr 26 2016
G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
T(2n,n) - T(2n,n+1) = n+1.

A118052 Number of partitions of n which contain their signature as a subpartition.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 8, 10, 16, 22, 32, 42, 58, 75, 101, 131, 174, 223, 293, 372, 480, 607, 772, 968, 1220, 1517, 1895, 2345, 2908, 3576, 4408, 5390, 6604, 8038, 9788, 11853, 14366, 17315, 20881, 25070, 30098, 35990, 43034, 51272, 61074, 72522, 86073, 101878, 120520

Offset: 0

Franklin T. Adams-Watters, Apr 10 2006

nonn

What is lim_{n->infinity} a(n)/p(n) (where p(n) = A000041(n) is the partition function)? It appears to be converging to something close to 0.8.

The limit must be at least 0.83846 = a(64)/p(64) and is probably closer to 0.9. - Charlie Neder, Aug 30 2018

			For n=3, signature([3]) = [1] is a subpartition of [3], signature([2,1]) = [1^2] is a subpartition of [2,1], but signature([1^3]) = [3] is not a subpartition of [1^3], so a(3)=2.

Charlie Neder, Table of n, a(n) for n = 0..64

Cf. A000041, A115621, A115622, A000041, A118053, A118054.

a(26) onwards from Charlie Neder, Aug 30 2018

A118053 Number of partitions of n which do not contain their signature as a subpartition.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 6, 8, 10, 14, 19, 26, 34, 45, 57, 74, 92, 118, 147, 185, 230, 287, 355, 441

Offset: 0

Franklin T. Adams-Watters, Apr 10 2006

			For n=3, signature([3]) = [1] is a subpartition of [3], signature([2,1]) = [1^2] is a subpartition of [2,1], but signature([1^3]) = [3] is not a subpartition of [1^3], so a(3)=1.

Cf. A118052, A115621, A115622, A000041.

cp's OEIS Frontend

A115623 Irregular triangle read by rows: row n lists numbers of distinct parts of partitions of n in Mathematica order.

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A115621 Signature of partitions in Abramowitz and Stegun order.

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A115262 Correlation triangle for n+1.

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A118052 Number of partitions of n which contain their signature as a subpartition.

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A118053 Number of partitions of n which do not contain their signature as a subpartition.

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