A100883
Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.
Original entry on oeis.org
1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305
Offset: 0
a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.
-
b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
`if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60); # Alois P. Heinz, Jul 03 2014
-
b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)
A006951
Number of conjugacy classes in GL(n,2).
Original entry on oeis.org
1, 1, 3, 6, 14, 27, 60, 117, 246, 490, 1002, 1998, 4053, 8088, 16284, 32559, 65330, 130626, 261726, 523374, 1047690, 2095314, 4192479, 8384808, 16773552, 33546736, 67101273, 134202258, 268420086, 536839446, 1073710914, 2147420250, 4294904430, 8589807438
Offset: 0
For the 5 partitions of 4 (namely [1^4]; [2,1^2]; [2^2]; [3,1]; [4]) we have
(f(m) = 2^(m-1)*(2-1) = 2^(m-1) and)
f([1^4]) = 2^3 = 8,
f([2,1^2]) = 1*2^1 = 2,
f([2^2]) = 2^1 = 2,
f([3,1]) = 1*1 = 1,
f([4]) = 1,
the sum is 8+2+2+1+1 = 14 = a(4).
- _Joerg Arndt_, Jan 02 2013
- W. D. Smith, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 161
- I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).
- N. J. A. Sloane, Transforms
-
/* The program does not work for n>19: */
[1] cat [NumberOfClasses(GL(n,2)): n in [1..19]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi Jan 24 2013
-
with(numtheory):
b:= n-> add(phi(d)*2^(n/d), d=divisors(n))/n-1:
a:= proc(n) option remember; `if`(n=0, 1,
add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Oct 20 2012
-
b[n_] := Sum[EulerPhi[d]*2^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Sum[2^(Length[ptn]-Length[Split[ptn]]),{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 21 2019 *)
-
N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-2*x^n) );
v=Vec(gf)
/* Joerg Arndt, Jan 02 2013 */
A075900
Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).
Original entry on oeis.org
1, 1, 3, 7, 19, 43, 115, 259, 659, 1523, 3731, 8531, 20883, 47379, 113043, 259219, 609683, 1385363, 3245459, 7344531, 17028499, 38579603, 88585619, 199845267, 457864595, 1028904339, 2339763603, 5256820115, 11896157587, 26626389395
Offset: 0
From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 19 splittings:
() (1) (2) (3) (4)
(1,1) (1,2) (1,3)
(1),(1) (2,1) (2,2)
(1,1,1) (3,1)
(2),(1) (1,1,2)
(1,1),(1) (1,2,1)
(1),(1),(1) (2,1,1)
(2),(2)
(3),(1)
(1,1,1,1)
(1,1),(2)
(1,2),(1)
(2),(1,1)
(2,1),(1)
(1,1),(1,1)
(1,1,1),(1)
(2),(1),(1)
(1,1),(1),(1)
(1),(1),(1),(1)
(End)
Partitions of partitions are
A001970.
Splittings with equal sums are
A074854.
Splittings of compositions are
A133494.
Splittings of partitions are
A323583.
Splittings with distinct sums are
A336127.
Starting with a reversed partition gives
A316245.
Starting with a partition instead of composition gives
A336136.
-
m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(&*[1-2^(j-1)*x^j: j in [1..m+2]]) )); // G. C. Greubel, Jan 25 2024
-
oo := 101; t1 := mul(1/(1-x^n/2),n=1..oo): t2 := series(t1,x,oo-1): t3 := seriestolist(t2): A075900 := n->2^n*t3[n+1];
with(combinat); A075900 := proc(n) local i,t1,t2,t3; t1 := partition(n); t2 := 0; for i from 1 to nops(t1) do t3 := t1[i]; t2 := t2+2^(n-nops(t3)); od: t2; end;
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b[n_]:= b[n]= Sum[d*2^(n - n/d), {d, Divisors[n]}];
a[0]= 1; a[n_]:= a[n]= 1/n*Sum[b[k]*a[n-k], {k,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Mar 20 2014, after Vladeta Jovovic, fixed by Vaclav Kotesovec, Mar 08 2018 *)
-
s(m,n):=if nVladimir Kruchinin, Sep 06 2014 */
-
{a(n)=polcoeff(prod(k=1,n,1/(1-2^(k-1)*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Jan 13 2013
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{a(n)=polcoeff(exp(sum(k=1,n+1,x^k/(k*(1-2^k*x^k)+x*O(x^n)))),n)} \\ Paul D. Hanna, Jan 13 2013
-
m=80;
def A075900_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( 1/product(1-2^(j-1)*x^j for j in range(1,m+1)) ).list()
A075900_list(m) # G. C. Greubel, Jan 25 2024
A304961
Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).
Original entry on oeis.org
1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832
Offset: 0
From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
() (1) (2) (3) (4)
(1,1) (1,2) (1,3)
(2,1) (2,2)
(1,1,1) (3,1)
(2),(1) (1,1,2)
(1,1),(1) (1,2,1)
(2,1,1)
(3),(1)
(1,1,1,1)
(1,2),(1)
(2,1),(1)
(1,1,1),(1)
(End)
Starting with a reversed partition gives
A323583.
Starting with a partition gives
A336134.
Partitions of partitions are
A001970.
Splittings with equal sums are
A074854.
Splittings of compositions are
A133494.
Splittings with distinct sums are
A336127.
-
nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020
A323583
Number of ways to split an integer partition of n into consecutive subsequences.
Original entry on oeis.org
1, 1, 3, 7, 17, 37, 83, 175, 373, 773, 1603, 3275, 6693, 13557, 27447, 55315, 111397, 223769, 449287, 900795, 1805465, 3615929, 7240327, 14491623, 29001625, 58027017, 116093259, 232237583, 464558201, 929224589, 1858623819, 3717475031, 7435314013, 14871103069
Offset: 0
The a(3) = 7 ways to split an integer partition of 3 into consecutive subsequences are (3), (21), (2)(1), (111), (11)(1), (1)(11), (1)(1)(1).
-
b:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i<1, 0,
b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
end:
a:= n-> ceil(b(n$2)):
seq(a(n), n=0..33); # Alois P. Heinz, Jan 01 2023
-
Table[Sum[2^(Length[ptn]-1),{ptn,IntegerPartitions[n]}],{n,40}]
(* Second program: *)
(1/2) CoefficientList[1 - 1/QPochhammer[2, x] + O[x]^100 , x] (* Jean-François Alcover, Jan 02 2022, after Vladimir Reshetnikov in A070933 *)
A229915
Number of espalier polycubes of a given volume in dimension 3.
Original entry on oeis.org
1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..100
- C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013.
- C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4.
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