Original entry on oeis.org
1, 1, 2, 2, 3, 3, 2, 1, 4, 5, 3, 1, 5, 7, 2, 1, 4, 2, 3, 1, 6, 11, 3, 1, 5, 2, 4, 1, 7, 15, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 22, 3, 1, 5, 2, 4, 1, 7, 4, 3, 1, 6, 2, 5, 1, 9, 30, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 7, 4, 1, 7, 2, 6, 1, 5, 1, 10, 42
Offset: 1
Written as an irregular triangle in which row n has length 2*A187219(n) we can see that the right border gives A000041 and the previous term of the last term in row n is n.
1,1;
2,2;
3,3;
2,1,4,5;
3,1,5,7;
2,1,4,2,3,1,6,11;
3,1,5,2,4,1,7,15;
2,1,4,2,3,1,6,4,5,1,4,1,8,22;
3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;
2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;
.
Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:
. _ _ _ _ _ _ _
. 15 _ _ _ _ |
. _ _ _ _|_ |
. _ _ _ | |
. _ _ _|_ _|_ |
. 11 _ _ _ | |
. _ _ _|_ | |
. _ _ | | |
. _ _|_ _|_ | |
. 7 _ _ _ | | |
. _ _ _|_ | | |
. 5 _ _ | | | |
. _ _|_ | | | |
. 3 _ _ | | | | |
. 2 _ | | | | | |
. 1 | | | | | | |
.
. 1 2 3 4 5 6 7
.
Also using the elements of this diagram we can draw a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See below:
.
7..................................
. /\
5.................... / \ /\
. /\ / \ /\ /
3.......... / \ / \ / \/
2..... /\ / \ /\/ \ /
1.. /\ / \ /\/ \ / \ /\/
0 /\/ \/ \/ \/ \/
. 0,2, 6, 12, 24, 40... = A211978
. 1, 4, 9, 19, 33... = A179862
.
Cf.
A000041,
A006128,
A135010,
A138137,
A141285,
A179862,
A186114,
A186412,
A187219,
A194446,
A206437,
A211978,
A220517,
A225600,
A225610.
A225610
Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.
Original entry on oeis.org
1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470
Offset: 0
For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.
. Diagram of regions
Partitions of 7 and partitions of 7
. _ _ _ _ _ _ _
7 15 |_ _ _ _ |
4 + 3 |_ _ _ _|_ |
5 + 2 |_ _ _ | |
3 + 2 + 2 |_ _ _|_ _|_ |
6 + 1 11 |_ _ _ | |
3 + 3 + 1 |_ _ _|_ | |
4 + 2 + 1 |_ _ | | |
2 + 2 + 2 + 1 |_ _|_ _|_ | |
5 + 1 + 1 7 |_ _ _ | | |
3 + 2 + 1 + 1 |_ _ _|_ | | |
4 + 1 + 1 + 1 5 |_ _ | | | |
2 + 2 + 1 + 1 + 1 |_ _|_ | | | |
3 + 1 + 1 + 1 + 1 3 |_ _ | | | | |
2 + 1 + 1 + 1 + 1 + 1 2 |_ | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1 1 |_|_|_|_|_|_|_|
.
. 1 2 3 4 5 6 7
.
Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:
. _ _ _ _ _ _
. |_ _ _ |
. |_ _ _|_ |
. |_ _ | |
. _ _ _ _ _ |_ _|_ _|_ |
. |_ _ _ | |_ _ _ | |
. _ _ _ _ |_ _ _|_ | |_ _ _|_ | |
. |_ _ | |_ _ | | |_ _ | | |
. _ _ _ |_ _|_ | |_ _|_ | | |_ _|_ | | |
. _ _ |_ _ | |_ _ | | |_ _ | | | |_ _ | | | |
. _ |_ | |_ | | |_ | | | |_ | | | | |_ | | | | |
.|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4 10 18 33 52 87
Cf.
A000041,
A000094,
A006128,
A066186,
A093694,
A133041,
A135010,
A138137,
A139250,
A139582,
A141285,
A182377,
A186114,
A186412,
A187219,
A194446,
A194447,
A206437,
A207779,
A211978,
A220517,
A225596,
A225600.
A194439
Number of regions in the set of partitions of n that contain only one part.
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297
Offset: 1
For n = 5 the seven regions of 5 in nondecreasing order are the sets of positive integers of the rows as shown below:
1;
1, 2;
1, 1, 3;
0, 0, 0, 2;
1, 1, 1, 2, 4;
0, 0, 0, 0, 0, 3;
1, 1, 1, 1, 1, 2, 5;
...
There are three regions that contain only one positive part, so a(5) = 3.
Note that in every column of the triangle the positive integers are also the parts of one of the partitions of 5.
Cf.
A000041,
A002865,
A027336,
A135010,
A138121,
A186114,
A186412,
A193870,
A194436,
A194437,
A194446,
A194447,
A206437.
A000094
Number of trees of diameter 4.
Original entry on oeis.org
0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 65, 88, 121, 161, 215, 280, 367, 471, 607, 771, 980, 1232, 1551, 1933, 2410, 2983, 3690, 4536, 5574, 6811, 8317, 10110, 12276, 14848, 17941, 21600, 25977, 31146, 37298, 44542, 53132, 63218, 75131, 89089
Offset: 1
From _Gus Wiseman_, Apr 12 2019: (Start)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts of size 2 or larger, or non-hooks, are the following. The Heinz numbers of these partitions are given by A105441.
(22) (32) (33) (43) (44)
(221) (42) (52) (53)
(222) (322) (62)
(321) (331) (332)
(2211) (421) (422)
(2221) (431)
(3211) (521)
(22111) (2222)
(3221)
(3311)
(4211)
(22211)
(32111)
(221111)
The a(5) = 1 through a(9) = 14 partitions of n-1 whose maximum part minus minimum part is at least 2 are the following. The Heinz numbers of these partitions are given by A307516.
(31) (41) (42) (52) (53)
(311) (51) (61) (62)
(321) (331) (71)
(411) (421) (422)
(3111) (511) (431)
(3211) (521)
(4111) (611)
(31111) (3221)
(3311)
(4211)
(5111)
(32111)
(41111)
(311111)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts that are smaller than the largest part are the following. The Heinz numbers of these partitions are given by A307517.
(211) (311) (321) (322) (422)
(2111) (411) (421) (431)
(2211) (511) (521)
(3111) (3211) (611)
(21111) (4111) (3221)
(22111) (3311)
(31111) (4211)
(211111) (5111)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(End)
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Christian G. Bower, Table of n, a(n) for n = 1..500
- J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
- J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
- Miloslav Znojil, Perturbation theory near degenerate exceptional points, arXiv:2008.00479 [math-ph], 2020.
- Index entries for sequences related to trees
-
g:=x/product(1-x^j,j=1..70)-x-x^2/(1-x)^2: gser:=series(g,x=0,48): seq(coeff(gser,x,n),n=1..46); # Emeric Deutsch, May 01 2006
A000094 := proc(n)
combinat[numbpart](n-1)-n+1 ;
end proc: # R. J. Mathar, May 17 2016
-
t=Table[PartitionsP[n]-n,{n,0,45}];
ReplacePart[t,0,1]
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[1/QPochhammer[x]-x/(1-x)^2-1+O[x]^50, x] (* Jean-François Alcover, Feb 04 2016 *)
A207779
Largest part plus the number of parts of the n-th region of the section model of partitions.
Original entry on oeis.org
2, 4, 6, 3, 9, 4, 12, 3, 6, 4, 17, 4, 7, 5, 22, 3, 6, 4, 10, 6, 5, 30, 4, 7, 5, 11, 4, 8, 6, 39, 3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 52, 4, 7, 5, 11, 4, 8, 6, 17, 6, 5, 11, 8, 7, 67, 3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 22, 4, 8, 6, 13, 5, 10, 8
Offset: 1
Written as a triangle begins:
2;
4;
6;
3, 9;
4, 12;
3, 6, 4, 17;
4, 7, 5, 22;
3, 6, 4, 10, 6, 5, 30;
4, 7, 5, 11, 4, 8, 6, 39;
3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 52;
Cf.
A002865,
A135010,
A182699,
A182709,
A183152,
A194436,
A194437,
A194438,
A194439,
A194447,
A206437.
A058399
Triangle of partial row sums of partition triangle A008284.
Original entry on oeis.org
1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1, 11, 10, 7, 4, 2, 1, 15, 14, 11, 7, 4, 2, 1, 22, 21, 17, 12, 7, 4, 2, 1, 30, 29, 25, 18, 12, 7, 4, 2, 1, 42, 41, 36, 28, 19, 12, 7, 4, 2, 1, 56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1, 77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1, 101, 100, 94, 80, 62
Offset: 1
From _Omar E. Pol_, Mar 10 2012: (Start)
Triangle begins:
1;
2, 1;
3, 2, 1;
5, 4, 2, 1;
7, 6, 4, 2, 1;
11, 10, 7, 4, 2, 1;
15, 14, 11, 7, 4, 2, 1;
22, 21, 17, 12, 7, 4, 2, 1;
30, 29, 25, 18, 12, 7, 4, 2, 1;
42, 41, 36, 28, 19, 12, 7, 4, 2, 1;
56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1;
77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1;
(End)
-
b:= proc(n, k) option remember;
`if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
end:
T:= (n, m)-> b(n,n) -b(n,m-1):
seq (seq (T(n, m), m=1..n), n=1..15); # Alois P. Heinz, Apr 20 2012
-
t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
A211009
Triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of j in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j), 1<=k<=A141285(n).
Original entry on oeis.org
1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 11, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 15, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 7, 22
Offset: 1
The irregular triangle begins:
1;
1, 2;
1, 1, 3;
1, 1;
1, 1, 2, 5;
1, 1, 1;
1, 1, 1, 2, 7;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 11;
1, 1, 1;
1, 1, 1, 2, 2;
1, 1, 1, 1;
1, 1, 1, 1, 2, 4, 15;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 4;
1, 1, 1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 2, 3, 7, 22;
...
From _Omar E. Pol_, Feb 06 2014: (Start)
Illustration of initial terms:
. _
. |_|
. 1
. _
. _|_|
. |_ _|
. 1 2
. _
. |_|
. _ _|_|
. |_ _ _|
. 1 1 3
. _ _
. |_ _|
. 1 1
. _
. |_|
. |_|
. _|_|
. _ _|_ _|
. |_ _ _ _|
. 1 1 2 5
.
(End)
A194438
Triangle read by rows: T(n,k) is the number of regions of the set of partitions of n into k parts.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 3, 1, 1, 0, 1, 0, 1, 5, 2, 1, 0, 1, 0, 1, 0, 0, 0, 1, 7, 3, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 11, 4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 15, 6, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0
Offset: 1
Triangle begins:
1;
1,1;
1,1,1;
2,1,1,0,1;
3,1,1,0,1,0,1;
5,2,1,0,1,0,1,0,0,0,1;
7,3,1,0,1,0,1,0,0,0,1,0,0,0,1;
11,4,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1;
...
Cf.
A008284,
A135010,
A138121,
A186114,
A186412,
A193870,
A194436,
A194437,
A194439,
A194446,
A194447,
A206437.
A182181
Total number of parts in the section model of partitions of A135010 with n regions.
Original entry on oeis.org
1, 3, 6, 7, 12, 13, 20, 21, 23, 24, 35, 36, 38, 39, 54, 55, 57, 58, 62, 63, 64, 86, 87, 89, 90, 94, 95, 97, 98, 128, 129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192, 193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275
Offset: 1
The first four regions of the section model of partitions are [1],[2, 1],[3, 1, 1],[2]. We can see that there are seven parts so a(4) = 7.
Written as a triangle begins:
1;
3;
6;
7, 12;
13, 20;
21, 23, 24, 35;
36, 38, 39, 54;
55, 57, 58, 62, 63, 64, 86;
87, 89, 90, 94, 95, 97, 98, 128;
129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192;
193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275;
...
From _Omar E. Pol_, Oct 20 2014: (Start)
Illustration of initial terms:
. _ _ _ _ _
. _ _ _ |_ _ _ |
. _ _ _ _ |_ _ _|_ |_ _ _|_ |
. _ _ |_ _ | |_ _ | |_ _ | |
. _ _ _ |_ _|_ |_ _|_ | |_ _|_ | |_ _|_ | |
. _ _ |_ _ | |_ _ | |_ _ | | |_ _ | | |_ _ | | |
. _ |_ | |_ | | |_ | | |_ | | | |_ | | | |_ | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
. 1 3 6 7 12 13 20
.
. _ _ _ _ _ _
. _ _ _ |_ _ _ |
. _ _ _ _ |_ _ _|_ |_ _ _|_ |
. _ _ |_ _ | |_ _ | |_ _ | |
. |_ _|_ _ _ |_ _|_ _|_ |_ _|_ _|_ |_ _|_ _|_ |
. |_ _ _ | |_ _ _ | |_ _ _ | |_ _ _ | |
. |_ _ _|_ | |_ _ _|_ | |_ _ _|_ | |_ _ _|_ | |
. |_ _ | | |_ _ | | |_ _ | | |_ _ | | |
. |_ _|_ | | |_ _|_ | | |_ _|_ | | |_ _|_ | | |
. |_ _ | | | |_ _ | | | |_ _ | | | |_ _ | | | |
. |_ | | | | |_ | | | | |_ | | | | |_ | | | | |
. |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 21 23 24 35
(End)
For the definition of "region" see
A206437.
-
lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; l = {};
For[j = 1, j <= 56, j++,
mx = Max@lex[j][[j]]; AppendTo[l, mx];
For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
AppendTo[reg, j - i];
];
Accumulate@reg (* Robert Price, Apr 22 2020, revised Jul 25 2020 *)
A207380
Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.
Original entry on oeis.org
0, 3, 10, 21, 42, 70, 122, 187, 298, 443, 667, 957, 1401, 1960, 2775, 3828, 5295, 7167, 9745, 12998, 17380, 22915, 30196, 39347, 51274, 66126, 85209, 108942, 139055, 176273, 223148, 280733, 352623, 440646, 549597, 682411, 845852, 1044084, 1286512, 1579582
Offset: 0
For n = 5 the three views of the three-dimensional shell model of partitions with 5 shells look like this:
.
. A066186(5) = 35 A006128(5) = 20
.
. 1 1 1 1 1 5
. 1 1 1 1 1 3 2
. 1 1 1 1 1 4 1
. 1 1 1 1 1 2 2 1
. 1 1 1 1 1 3 1 1
. 1 1 1 1 1 2 1 1 1
. 1 1 1 1 1 1 1 1 1 1
.
.
. 7 6 4 2 1
. 1 2 3 2
. 1 1 2
. 1 1
. 1
.
. A000217(5) = 15
.
The areas of the shadows of the three views are A066186(5) = 35, A006128(5) = 20 and A000217(5) = 15, therefore the total area of the three shadows is 35+20+15 = 70, so a(5) = 70.
Cf.
A000041,
A000217,
A006128,
A026792,
A066186,
A135010,
A138121,
A141285,
A182703,
A182715,
A206437,
A209655.
-
b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, n]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+g[1]]
fi
end:
a:= n-> n*b(n, n)[1] +b(n, n)[2] +n*(n+1)/2:
seq (a(n), n=0..50); # Alois P. Heinz, Mar 22 2012
-
b[n_, i_] := b[n, i] = Module[{f, g}, If [n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; Join[f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]] ]]]; a[n_] := n*b[n, n][[1]] + b[n, n][[2]] + n*(n+1)/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *)
Comments