A330661 -id:A330661 - cp's OEIS frontend

A238640 Position of [n, n, ..., n] (n n's) in Mathematica-ordered list of partitions of n^2.

1, 1, 3, 19, 168, 1582, 15546, 157051, 1625368, 17159223, 184277224, 2008388660, 22172275440, 247558926150, 2791793968821, 31764451979736, 364283594455091, 4207485803818522, 48908343969469479, 571811846280602486, 6720473048598172508, 79363083519870386700

Offset: 0

Clark Kimberling, Mar 04 2014

nonn

			The partitions of 4 in Mathematica order are 4, 31, 22, 211, 1111.  The position of 22 is a(2) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..140

Cf. A000290, A072213, A080577 (Mathematica ordering), A238638, A238639, A330661, A332706.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> 1 +add(b(n^2-j, j), j=n+1..n^2):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 03 2014

r[n_] := Table[n, {k, 1, n}]; Flatten[Table[Position[IntegerPartitions[n^2], r[n]], {n, 0, 8}]]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, i]]]]; a[n_] := 1+Sum[b[n^2-j, j], {j, n+1, n^2}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

a(9)-a(21) from Alois P. Heinz, Sep 03 2014

A332706 Index position of {2}^n within the list of partitions of 2n in canonical ordering.

Original entry on oeis.org

1, 1, 3, 8, 18, 37, 71, 128, 223, 376, 617, 991, 1563, 2423, 3704, 5589, 8333, 12293, 17959, 25996, 37318, 53153, 75153, 105535, 147249, 204201, 281563, 386128, 526795, 715191, 966437, 1300125, 1741598, 2323487, 3087701, 4087933, 5392747, 7089463, 9289053

Offset: 0

Alois P. Heinz, Feb 20 2020

nonn

The canonical ordering of partitions is described in A080577.

			a(3) = 8, because 222 has position 8 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Bisection (even part) of A058984.

Cf. A000041, A080577, A216053, A238639, A238640, A322761, A330661.

a:= n-> combinat[numbpart](2*n)-n:
seq(a(n), n=0..44);

a[n_] := PartitionsP[2n] - n;
Table[a[n], {n, 0, 44}] (* Jean-François Alcover, Aug 20 2021, from Maple *)

a(n) = A000041(2n) - n.
a(n) = A058984(2n).
a(n) = A330661(2n,n).

A238639 Position of [n, n-1, ..., 2, 1] in Mathematica-ordered list of partitions of n(n+1)/2.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 498, 2493, 12741, 66224, 348963, 1859009, 9994196, 54155387, 295477841, 1621962199, 8951635343, 49644856801, 276540258555, 1546630084062, 8681889729354, 48900895532763, 276302483274825, 1565747892473958, 8896975706929141, 50683901455201860

Offset: 0

Clark Kimberling, Mar 04 2014

nonn

			The partitions of 6 in Mathematica order are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.  The position of 321 is a(3) = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Manfred Scheucher, C Code

Cf. A000217, A080577 (Mathematica ordering), A238638, A238640, A330661, A332706.

g:= (n, i)-> `if`(n=0, 1, g(n-i+1, i-1)+ b(n-i, i)):
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> (m-> add(b(m-j, min(j, m-j)), j=n+1..m)+
                 g(m-n, n))(n*(n+1)/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 03 2015

r[n_] := Table[n - k, {k, 0, n - 1}]; Flatten[Table[Position[IntegerPartitions[n (n + 1)/2], r[n]], {n, 0, 2}]]
g[n_, i_] := If[n==0, 1, g[n-i+1, i-1] + b[n-i, i]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Function[m, Sum[b[m-j, Min[j, m-j]], {j, n+1, m}] + g[m-n, n]][n(n+1)/2]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

a(13)-a(17) from Manfred Scheucher, Jun 01 2015

a(18)-a(25) from Alois P. Heinz, Jun 02 2015

A216053 a(n) is the position of the last two-tuple within the reverse lexicographic set of partitions of 2n and 2n+1, with a(1)-a(n) representing the positions of every 2-tuple partition of 2n and 2n+1.

Original entry on oeis.org

2, 3, 5, 8, 13, 20, 31, 46, 68, 98, 140, 196, 273, 374, 509, 685, 916, 1213, 1598, 2088, 2715, 3507, 4509, 5764, 7339, 9297, 11733, 14743, 18461, 23026, 28630, 35472, 43821, 53964, 66274, 81157, 99134, 120771, 146786, 177971, 215309, 259892, 313066, 376327

Offset: 1

J. Stauduhar, Oct 12 2012

nonn

			With n = 3, 2n = 6.  The partitions of 6 are {{6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1,1}, {1,1,1,1,1,1}}.  The last 2-tuple is located at position 5. The positions of all 2-tuples are 2, 3, and 5.

J. Stauduhar, Table of n, a(n) for n = 1..10000

A diagonal of A181187.

Cf. A080577, A330661, A332706.

RecurrenceTable[{a[n+1] == a[n] + PartitionsP[(n)], a[1] == 2}, a, {n, 1, 44}]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, May 24 2018

a(n) = A330661(2n,2) = A330661(2n+1,2). - Alois P. Heinz, Feb 20 2020

A332719 Index position of {n}^3 within the list of partitions of 3n in canonical ordering.

Original entry on oeis.org

1, 3, 8, 19, 44, 93, 187, 357, 657, 1166, 2015, 3393, 5594, 9044, 14378, 22501, 34734, 52931, 79735, 118823, 175337, 256347, 371606, 534377, 762721, 1080979, 1521925, 2129330, 2961580, 4096006, 5634855, 7712558, 10505457, 14243772, 19227383, 25845241, 34600673

Offset: 0

Alois P. Heinz, Feb 20 2020

nonn

The canonical ordering of partitions is described in A080577.

			a(2) = 8, because 222 has position 8 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..4000
Wikipedia, Integer Partition

Cf. A000041, A080577, A322761, A330661, A332706, A332720.

b:= proc(n, i) option remember;
     `if`(n=0, 1, b(n-i, i)+g(n, i-1))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
a:= n-> g(3*n$2)-b(3*n, n)+1:
seq(a(n), n=0..37);

b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i] + g[n, i - 1]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
a[n_] := g[3n, 3n] - b[3n, n] + 1;
a /@ Range[0, 37] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*sqrt(n)). - Vaclav Kotesovec, Feb 28 2020

A332720 Index position of {3}^n within the list of partitions of 3n in canonical ordering.

Original entry on oeis.org

1, 1, 5, 19, 59, 150, 349, 745, 1515, 2936, 5514, 10036, 17851, 31039, 53006, 88943, 147057, 239701, 385885, 613855, 966137, 1505137, 2323124, 3553914, 5392315, 8117758, 12131618, 18003740, 26543030, 38886999, 56633453, 82009410, 118113488, 169229009, 241264461

Offset: 0

Alois P. Heinz, Feb 20 2020

nonn

The canonical ordering of partitions is described in A080577.

			a(2) = 5, because 33 has position 5 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..4000
Wikipedia, Integer Partition

Cf. A000041, A080577, A322761, A330661, A332706, A332719.

b:= proc(n) option remember;
     `if`(n=0, 1, b(n-1)+g(3*n, 2))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
a:= n-> g(3*n$2)-b(n)+1:
seq(a(n), n=0..35);

b[n_] := b[n] = If[n == 0, 1, b[n - 1] + g[3n, 2]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
a[n_] := g[3n, 3n] - b[n] + 1;
a /@ Range[0, 35] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(2*n)) / (4*3^(3/2)*n). - Vaclav Kotesovec, Feb 28 2020

A330693 Regular subtriangle of A328773: T(n,k) is the number of the balanced colored digraphs on n nodes with k colors, 1 <= k <= n.

Original entry on oeis.org

1, 3, 4, 16, 36, 64, 218, 1104, 2112, 4096, 9608, 90416, 266496, 528384, 1048576, 1540944, 30194176, 135032832, 269500416, 537919488, 1073741824

Offset: 1

Peter Dolland, Dec 25 2019

The color count, k, is a measure of the differentiation of the nodes: k = 1 means no differentiation at all, so we have A000273. k = n means all nodes are taken to be different, and we have A053763. With increasing differentiation T(n,k) < T(n,k+1) for 1 <= k < n.

			The balanced color scheme [2,2,2] has with A330661 the index 5 in A328773. So there are T(6,3) = A328773(6,5) = 135032832 digraphs with 6 nodes two each in the same color.
Triangle T(n,k) begins:
        1;
        3,        4;
       16,       36,        64;
      218,     1104,      2112,      4096;
     9608,    90416,    266496,    528384,   1048576;
  1540944, 30194176, 135032832, 269500416, 537919488, 1073741824;
  ...

Cf. A328773, A330661, A000273, A053763.

\\ here C(p) computes sequence value for given partition.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
C(p)={((i, v)->if(i>#p, 2^edges(v), my(s=0); forpart(q=p[i], s+=permcount(q)*self()(i+1, concat(v, Vec(q)))); s/p[i]!))(1, [])}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n),p[#p]-p[1]<=1], , 4))}
{ for(n=1, 6, print(Row(n))) }

T(n,1) = A000273(n).
T(n,n) = A053763(n) = 2^(n^2 - n).
T(n,k) = A328773(n,A330661(n,k)).

cp's OEIS Frontend

A238640 Position of [n, n, ..., n] (n n's) in Mathematica-ordered list of partitions of n^2.

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A332706 Index position of {2}^n within the list of partitions of 2n in canonical ordering.

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A238639 Position of [n, n-1, ..., 2, 1] in Mathematica-ordered list of partitions of n(n+1)/2.

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A216053 a(n) is the position of the last two-tuple within the reverse lexicographic set of partitions of 2n and 2n+1, with a(1)-a(n) representing the positions of every 2-tuple partition of 2n and 2n+1.

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A332719 Index position of {n}^3 within the list of partitions of 3n in canonical ordering.

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A332720 Index position of {3}^n within the list of partitions of 3n in canonical ordering.

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A330693 Regular subtriangle of A328773: T(n,k) is the number of the balanced colored digraphs on n nodes with k colors, 1 <= k <= n.

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