A000065 -id:A000065 - cp's OEIS frontend

A263294 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n vertices and treewidth k.

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 9, 17, 6, 1, 1, 19, 72, 53, 10, 1, 1, 36, 323, 501, 168, 14, 1, 1, 75, 1639, 5889, 4163, 557, 21, 1, 1, 152, 9203, 81786, 138923, 42596, 1977, 29, 1

Offset: 1

Christian Stump, Oct 13 2015

A graph without edges has treewidth 0, any other forest has treewidth 1, any other series parallel graph has treewidth 2. - Martin Rubey, May 10 2023

			Triangle begins:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    4,     1;
  1,   9,   17,     6,      1;
  1,  19,   72,    53,     10,     1;
  1,  36,  323,   501,    168,    14,    1;
  1,  75, 1639,  5889,   4163,   557,   21,  1;
  1, 152, 9203, 81786, 138923, 42596, 1977, 29, 1;
  ...

FindStat - Combinatorial Statistic Finder, The treewidth of a graph.
Wikipedia, Treewidth

Columns k=2..3 are A362908, A362907.
Partial row sums include A000012, A005195, A000041.
Row sums are A000088.
T(n,n-2) = A000065(n).

Corrected and extended by Martin Rubey, May 10 2023

A291336 Number F(n,h,t) of forests of t unlabeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 0, 4, 3, 1, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 3, 2, 1, 0, 6, 8, 3, 1, 0, 8, 4, 1, 0, 4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 4, 3, 2, 1, 0, 10, 15, 9, 3, 1, 0, 18, 13, 4, 1, 0, 13, 5, 1, 0, 5, 1, 0, 1, 0

Offset: 0

Alois P. Heinz, Aug 22 2017

Elements in rows h=0 give A023531.
Positive elements in rows h=1 give A008284.
Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A034781.
Positive column sums per layer give A033185.

			n h\t: 0 1 2 3 4 5 : A034781 : A033185   : A000081
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 1       :       1 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 1       : 2
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 1     :       1 : . . .     :
3 1  : 0 1 1       :       2 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 2 1 1     : 4
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 1   :       1 : . . . .   :
4 1  : 0 1 2 1     :       4 : . . .     :
4 2  : 0 2 1       :       3 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 4 3 1 1   : 9
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 1 :       1 : . . . . . :
5 1  : 0 1 2 2 1   :       6 : . . . .   :
5 2  : 0 4 3 1     :       8 : . . .     :
5 3  : 0 3 1       :       4 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 9 6 3 1 1 : 20
-----+-------------+---------+-----------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000041, A000065, A000081, A001853, A005197, A008284, A023531, A033185, A034781, A291203, A291204, A291529.

b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0
       or i=1, x^(t*n), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-1)+j-1, j)*b(n-i*j, i-1, t, h), j=1..n/i)))
    end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..9);

b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0
     || i == 1, x^(t*n), b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
     b[i-1, i-1, 0, h-1]+j-1, j]*b[n - i*j, i-1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h-1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n-h}], {h, 0, n}], {n, 0, 9}] //
Flatten (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000081(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A005197(n).
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A001853(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A000065(n) = A000041(n) - 1.
F(n,1,1) = 1 for n>1.
F(n,0,0) = A000007(n).

A347542 Number of partitions of n into 6 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 44, 65, 92, 130, 178, 244, 326, 435, 571, 747, 964, 1242, 1581, 2009, 2530, 3178, 3962, 4930, 6094, 7518, 9225, 11296, 13768, 16751, 20295, 24546, 29583, 35591, 42685, 51112, 61028, 72757, 86523, 102740, 121720, 144007, 170018, 200461, 235910, 277270

Offset: 6

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A001402, A004250, A035300, A035301, A347543, A347544, A347545, A347547.

nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 6, nmax}], {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: Sum_{k>=6} x^k / Product_{j=1..k} (1 - x^j).

A347543 Number of partitions of n into 7 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 134, 186, 255, 345, 461, 611, 801, 1043, 1346, 1727, 2199, 2787, 3508, 4398, 5482, 6809, 8414, 10365, 12711, 15545, 18935, 23006, 27854, 33646, 40513, 48680, 58326, 69748, 83192, 99048, 117650, 139513, 165083, 195034, 229968, 270760

Offset: 7

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008636, A035300, A035301, A347542, A347544, A347545, A347547.

nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 7, nmax}], {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: Sum_{k>=7} x^k / Product_{j=1..k} (1 - x^j).

A347544 Number of partitions of n into 8 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 96, 137, 190, 263, 356, 480, 637, 842, 1098, 1427, 1835, 2351, 2986, 3780, 4749, 5949, 7405, 9190, 11344, 13966, 17111, 20913, 25454, 30908, 37393, 45141, 54315, 65222, 78090, 93317, 111220, 132323, 157050, 186088, 220015, 259716

Offset: 8

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008637, A035300, A035301, A347542, A347543, A347545, A347547.

nmax = 52; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 8, nmax}], {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: Sum_{k>=8} x^k / Product_{j=1..k} (1 - x^j).

A347545 Number of partitions of n into 9 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 138, 193, 267, 364, 491, 656, 868, 1139, 1483, 1917, 2461, 3142, 3985, 5030, 6315, 7893, 9817, 12165, 15007, 18451, 22597, 27589, 33565, 40724, 49249, 59410, 71460, 85753, 102632, 122574, 146032, 173638, 206003, 243951, 288296, 340124

Offset: 9

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008638, A035300, A035301, A347542, A347543, A347544, A347547.

nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 9, nmax}], {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: Sum_{k>=9} x^k / Product_{j=1..k} (1 - x^j).

A347547 Number of partitions of n into 10 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 270, 368, 499, 667, 887, 1165, 1524, 1973, 2544, 3253, 4143, 5239, 6602, 8268, 10320, 12813, 15859, 19537, 24000, 29359, 35820, 43541, 52795, 63803, 76929, 92476, 110926, 132694, 158414, 188649, 224231, 265916, 314793

Offset: 10

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008639, A035300, A035301, A347542, A347543, A347544, A347545.

nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 10, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: Sum_{k>=10} x^k / Product_{j=1..k} (1 - x^j).

A167930 Number of partitions of n in which some but not all parts are equal.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 4, 9, 13, 20, 29, 43, 57, 82, 110, 146, 195, 258, 334, 435, 558, 713, 910, 1150, 1446, 1814, 2268, 2815, 3491, 4308, 5301, 6501, 7954, 9692, 11795, 14295, 17301, 20876, 25148, 30200, 36218, 43322, 51741, 61650, 73354

Offset: 0

Omar E. Pol, Nov 15 2009

nonn

The parts may not all be equal, and at least one part must occur at least twice. - N. J. A. Sloane, May 30 2024

			The partitions of 6 are:
6 ....................... All parts are distinct.
5 + 1 ................... All parts are distinct.
4 + 2 ................... All parts are distinct.
4 + 1 + 1 ............... Only some parts are equal ...... (1).
3 + 3 ................... All parts are equal.
3 + 2 + 1 ............... All parts are distinct.
3 + 1 + 1 + 1 ........... Only some parts are equal ...... (2).
2 + 2 + 2 ............... All parts are equal.
2 + 2 + 1 + 1 ........... Only some parts are equal ...... (3).
2 + 1 + 1 + 1 + 1 ....... Only some parts are equal ...... (4).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
Then a(6) = 4.
a(7) = 9 from 511  4111  331  322  3211  31111  2221  22111  211111. - _N. J. A. Sloane_, May 30 2024

Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167931, A167932, A167933.

f[lst_]:=With[{c=Split[lst]},Length[lst]>2&&Max[Length/@c]>1&&Length[c]>1]; Table[Length[ Select[ IntegerPartitions[n],f]],{n,0,50}] (* Harvey P. Dale, May 30 2024 *)

a(n) = A047967(n) - A032741(n).
a(n) = A000041(n) - A000009(n) - A032741(n).
a(0) = 0: For n>0, a(n) = A000041(n) - A000009(n) - A000005(n) + 1.

Edited by Omar E. Pol, Nov 16 2009

More terms from Max Alekseyev, May 02 2011

A167932 Number of partitions of n such that all parts are equal or all parts are distinct.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 6, 9, 10, 13, 13, 20, 19, 25, 30, 36, 39, 51, 55, 69, 79, 92, 105, 129, 144, 168, 195, 227, 257, 303, 341, 395, 451, 515, 588, 676, 761, 867, 985, 1120, 1261, 1433, 1611, 1821, 2053, 2307, 2591, 2919, 3266, 3663, 4100, 4587, 5121, 5725, 6381

Offset: 0

Omar E. Pol, Nov 15 2009

nonn

Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)).

			The partitions of 6 are:
6 .............. All parts are distinct ..... (1).
5+1 ............ All parts are distinct ..... (2).
4+2 ............ All parts are distinct ..... (3).
4+1+1 .......... Only some parts are equal.
3+3 ............ All parts are equal ........ (4).
3+2+1 .......... All parts are distinct ..... (5).
3+1+1+1 ........ Only some parts are equal.
2+2+2 .......... All parts are equal ........ (6).
2+2+1+1 ........ Only some parts are equal.
2+1+1+1+1 ...... Only some parts are equal.
1+1+1+1+1+1 .... All parts are equal ........ (7).
So a(6) = 7.

Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933.

ds[n_]:=Module[{lun=Length[Union[n]]},Length[n]==lun||lun==1]; Table[ Count[ IntegerPartitions[n],?(ds)],{n,0,60}] (* _Harvey P. Dale, Sep 13 2011 *)

a(n) = A000041(n) - A167930(n).

a(n) = A000009(n) + A032741(n).

More terms from D. S. McNeil, May 10 2010

A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 66, 90, 118, 156, 201, 264, 336, 434, 550, 700, 880, 1112, 1385, 1733, 2149, 2666, 3283, 4049, 4956, 6072, 7398, 9009, 10922, 13237, 15970, 19261, 23147, 27790, 33260, 39776, 47425, 56497, 67133, 79685, 94371, 111653

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - Gus Wiseman, Jul 10 2023

			a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
From _Gus Wiseman_, Jul 10 2023 (Start)
The a(2) = 1 through a(8) = 14 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)
       (21)  (22)  (32)   (33)    (43)    (44)
             (31)  (41)   (42)    (52)    (53)
                   (221)  (51)    (61)    (62)
                          (222)   (322)   (71)
                          (321)   (331)   (332)
                          (2211)  (421)   (422)
                                  (2221)  (431)
                                  (3211)  (521)
                                          (2222)
                                          (3221)
                                          (3311)
                                          (4211)
                                          (22211)
(End)

Cf. A096373.
For mean instead of median we have A000065, ranks A057716.
The complement is counted by A027336, ranks A364056.
Rows sums of A359893 if we remove the first column.
These partitions have ranks A364058.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A124943 counts partitions by low median, high A124944.
A241131 counts partitions with low mode 1, ranks A360015.
Cf. A000070, A000975, A002865, A110618, A237984, A363488.

Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* Gus Wiseman, Jul 10 2023 *)

From Gus Wiseman, Jul 11 2023: (Start)
a(n>2) = A000041(n) - A096373(n-2).
a(n>1) = A000041(n-2) + A002865(n+1).
a(n) = A000041(n+1) - A027336(n).
(End)

Formula corrected by Gus Wiseman, Jul 11 2023

cp's OEIS Frontend

A263294 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n vertices and treewidth k.

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A291336 Number F(n,h,t) of forests of t unlabeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

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Maple

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A347542 Number of partitions of n into 6 or more parts.

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Programs

Mathematica

Formula

A347543 Number of partitions of n into 7 or more parts.

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Keywords

Crossrefs

Programs

Mathematica

Formula

A347544 Number of partitions of n into 8 or more parts.

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Keywords

Crossrefs

Programs

Mathematica

Formula

A347545 Number of partitions of n into 9 or more parts.

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Programs

Mathematica

Formula

A347547 Number of partitions of n into 10 or more parts.

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Crossrefs

Programs

Mathematica

Formula

A167930 Number of partitions of n in which some but not all parts are equal.

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Mathematica

Formula

Extensions

A167932 Number of partitions of n such that all parts are equal or all parts are distinct.

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