A182983 -id:A182983 - cp's OEIS frontend

A182980 Version "mirror" of the shell model of partitions of A135010. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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

Offset: 1

Omar E. Pol, Feb 02 2011

In n is odd then row n lists the numbers of row n of A135010. If n is even the row n lists the 1's of row n of A135010 and then row n lists the other numbers of row n of A135010 in reverse order.

			Triangle begins:
1,
1, 2,
1, 1, 3,
1, 1, 1, 4, 2, 2,
1, 1, 1, 1, 1, 2, 3, 5,
1, 1, 1, 1, 1, 1, 1, 6, 3, 3, 2, 4, 2, 2, 2,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 5, 3, 4, 7,

Cf. A135010, A141285, A182742, A182743, A182982, A182983, A182985.

A194795 Imbalance of the number of partitions of n.

Original entry on oeis.org

0, -1, 0, -2, 0, -4, 0, -7, 1, -11, 3, -18, 6, -28, 13, -42, 24, -64, 41, -96, 69, -141, 112, -208, 175, -303, 271, -437, 410, -629, 609, -898, 896, -1271, 1302, -1792, 1868, -2510, 2660, -3493, 3752, -4839, 5248, -6666, 7293, -9131, 10065, -12454

Offset: 1

Omar E. Pol, Feb 02 2012

sign

Consider the three-dimensional structure of the shell model of partitions version "tree". Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns. Note that every column contains exactly the same parts, the same as a periodic table (see example). For more information see A135010.

			For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
------------------------------------------------------
Partitions                Tree             Table 1.0
of 6.                    A194805            A135010
------------------------------------------------------
6                   6                     6 . . . . .
3+3                   3                   3 . . 3 . .
4+2                     4                 4 . . . 2 .
2+2+2                     2               2 . 2 . 2 .
5+1                         1   5         5 . . . . 1
3+2+1                       1 3           3 . . 2 . 1
4+1+1                   4   1             4 . . . 1 1
2+2+1+1                   2 1             2 . 2 . 1 1
3+1+1+1                     1 3           3 . . 1 1 1
2+1+1+1+1                 2 1             2 . 1 1 1 1
1+1+1+1+1+1                 1             1 1 1 1 1 1
------------------------------------------------------
.
.                   6 3 4 2 1 3 5
.     Table 2.0     . . . . 1 . .     Table 2.1
.      A182982      . . . 2 1 . .      A182983
.                   . 3 . . 1 2 .
.                   . . 2 2 1 . .
.                   . . . . 1
------------------------------------------------------
The number of partitions with parts on the left hand side is equal to 7 and the number of partitions with parts on the right hand side is equal to 3, so a(6) = -7+3 = -4. On the other hand; for n = 6 the first n terms of A002865 (with positive indices) are 0, 1, 1, 2, 2, 4 therefore a(6) = 0-1+1-2+2-4 = -4.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A002865, A135010, A182703, A187219, A194796, A194797, A194805, A194809.

with(combinat):
a:= proc(n) option remember;
      (-1)^n *(numbpart(n-1)-numbpart(n)) +`if`(n>1, a(n-1), 0)
    end:
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012

a[n_] := a[n] = (-1)^n*(PartitionsP[n-1]-PartitionsP[n]) + If[n>1, a[n-1], 0]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
nmax = 60; Rest[CoefficientList[Series[x/(1-x) - (1+x)/(1-x) * Product[1/((1 + x^(2*k-1))*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2015 *)
nmax = 60; Rest[CoefficientList[Series[-x/(1+x) - (1-x)/(1+x) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2015 *)

a(n) = Sum_{k=1..n} (-1)^(k-1)*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
a(n) = Sum_{k=1..n} (-1)^(k-1)*A002865(k).
a(n) = (-1)^(n+1) * (A240690(n+1) - A240690(n)) - 1. - Vaclav Kotesovec, Nov 11 2015
a(n) ~ (-1)^(n+1) * Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Nov 11 2015

A182718 Largest part of the n-th row of the diagram of another version of the shell model of partitions in which each last section of partitions is represented by a tree.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 26 2011

The same positive integers of the triangle of A141285 but here each row lists the terms in reverse order.
It appears that the version of the shell model of partitions which gives this sequence also shows a connection between the partition theory and the graph theory because the structure looks like a forest or grove in which each last section is a tree. In a three-dimensional version of the structure we can see that one of the views shows the branches of the trees of the even numbers tilted 45 degrees to the left and the branches of the trees of the odd numbers tilted 45 degrees to the right. Another view contains the "roots" of trees, the tables of both A182982 and A182983.
I would like a table for this sequence!
I would like to see the graphic!

			Triangle begins:
1,
2,
3,
4, 2,
5, 3,
6, 3, 4, 2,
7, 4, 5, 3,
8, 4, 5, 6, 3, 4, 2,
9, 5, 6, 3, 7, 4, 5, 3,

Cf. A135010, A138121, A141285, A182742, A182743, A182982, A182983.

Column 1 gives A000027.

A182985 Largest part of the n-th row of the table version "mirror" of the shell model of partitions of A135010 and A182980.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 02 2011

If n is odd then row n lists the numbers of row n of A141285. If n is even then row n lists the numbers of row n of A141285 but in reverse order.

			Triangle begins:
1,
2,
3,
4, 2,
3, 5,
6, 3, 4, 2,
3, 5, 4, 7,
8, 4, 5, 6, 3, 4, 2,
3, 5, 4, 7, 3, 6, 5, 9,
10,5, 6, 7, 4, 8, 4, 5, 6, 3, 4, 2,
3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11,

Cf. A135010, A181285, A182982, A182983.

A182981 Number of grid point that are covered at the n-th stage of the cellular automata which is a version "forest" of the shell model of partitions, with a(0)=1. The outers shells are represented as trees.

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 31, 39

Offset: 0

Omar E. Pol, Jan 26 2011

nonn

It appears that the version of the shell model of partitions which gives this sequence also shows a connection between the partition theory and the graph theory because the structure looks like a forest in which each last section is a tree. In a three-dimensional version of the structure we can see that one of the views shows the branches of the trees of the even numbers tilted 45 degrees to the left and the branches of the trees of the odd numbers tilted 45 degrees to the right. Another view contains the "roots" of trees, the tables of both A182982 and A182983.

Cf. A135010, A182718, A182982, A182983.

cp's OEIS Frontend

A182980 Version "mirror" of the shell model of partitions of A135010. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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A194795 Imbalance of the number of partitions of n.

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A182718 Largest part of the n-th row of the diagram of another version of the shell model of partitions in which each last section of partitions is represented by a tree.

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A182985 Largest part of the n-th row of the table version "mirror" of the shell model of partitions of A135010 and A182980.

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A182981 Number of grid point that are covered at the n-th stage of the cellular automata which is a version "forest" of the shell model of partitions, with a(0)=1. The outers shells are represented as trees.

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