A210945
Triangle read by rows: T(n,k) = number of parts in the k-th column of the mirror of the last shell of the partitions of n.
Original entry on oeis.org
1, 2, 3, 5, 1, 7, 1, 11, 3, 1, 15, 3, 1, 22, 6, 3, 1, 30, 7, 4, 1, 42, 11, 7, 3, 1, 56, 13, 9, 4, 1, 77, 20, 15, 8, 3, 1, 101, 23, 18, 10, 4, 1, 135, 33, 27, 17, 8, 3, 1, 176, 40, 34, 22, 11, 4, 1, 231, 54, 47, 33, 18, 8, 3, 1, 297, 65, 58, 42, 24, 11, 4, 1
Offset: 1
For n = 7 the illustration shows two arrangements of the last shell of the partitions of 7:
.
. (7) (7)
. (4+3) (3+4)
. (5+2) (2+5)
. (3+2+2) (2+2+3)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. (1) (1)
. --------
. 15,3,1
.
We can see that in the right hand picture (the mirror) the number of part for columns 1..3 are 15, 3, 1 therefore row 7 lists 15, 3, 1.
Written as a triangle begins:
1;
2;
3;
5, 1;
7, 1;
11, 3, 1;
15, 3, 1;
22, 6, 3, 1;
30, 7, 4, 1;
42, 11, 7, 3, 1;
56, 13, 9, 4, 1;
77, 20, 15, 8, 3, 1;
101, 23, 18, 10, 4, 1;
135, 33, 27, 17, 8, 3, 1;
176, 40, 34, 22, 11, 4, 1;
231, 54, 47, 33, 18, 8, 3, 1;
297, 65, 58, 42, 24, 11, 4, 1;
A119907
Number of partitions of n such that if k is the largest part, then k-2 occurs as a part.
Original entry on oeis.org
0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 15, 18, 27, 34, 47, 58, 79, 96, 127, 155, 199, 242, 308, 371, 465, 561, 694, 833, 1024, 1223, 1491, 1778, 2150, 2556, 3076, 3642, 4359, 5151, 6133, 7225, 8570, 10066, 11892, 13937, 16401, 19173, 22495, 26228, 30676, 35692, 41620
Offset: 0
-
b:= proc(n, i) option remember;
`if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
end:
a:= n-> add(b(n-(2*k-2), k), k=3..1+n/2):
seq(a(n), n=0..60); # Alois P. Heinz, May 18 2012
-
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i>n, 0, b[n-i, i]]]; a[n_] := Sum[b[n-(2*k-2), k], {k, 3, 1+n/2}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)
A268190
Triangle read by rows: T(n,k) (n, k>=1) is the number of partitions of n such that the difference between the two largest distinct parts is k; T(n,0) is the number of partitions of n in which all parts are equal.
Original entry on oeis.org
1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 7, 5, 2, 2, 1, 1, 3, 11, 5, 5, 2, 2, 1, 1, 4, 13, 10, 5, 4, 2, 2, 1, 1, 2, 20, 11, 8, 5, 4, 2, 2, 1, 1, 6, 23, 16, 10, 8, 4, 4, 2, 2, 1, 1, 2, 33, 20, 15, 9, 8, 4, 4
Offset: 1
T(5,0)=2 because we have [5] and [1,1,1,1,1]; T(5,1)=3 because we have [3,2], [2,2,1], and [2,1,1,1]; T(5,2)=1 because we have [3,1,1]; T(5,3)=1 because we have [4,1].
Triangle starts:
1;
2;
2,1;
3,1,1;
2,3,1,1;
4,3,2,1,1;
-
G := add(x^k/(1-x^k), k = 1 .. 80)+ add(add(t^(i-j)*x^(i+j)/((1-x^i)*mul(1-x^k,k = 1 .. j)), j = 1 .. i-1), i = 2 .. 80): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: 1; for n from 2 to 25 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, l, i) option remember; `if`(irem(n, i)=0, x^
`if`(l=0, 0, i-l), 0) +`if`(i>n, 0, add(b(n-i*j,
`if`(j=0, l, i), i+1), j=0..(n-1)/i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=1..30); # Alois P. Heinz, Feb 11 2016
-
b[n_, l_, i_] := b[n, l, i] = If[Mod[n, i] == 0, x^If[l == 0, 0, i-l], 0] + If[i>n, 0, Sum[b[n-i*j, If[j == 0, l, i], i+1], {j, 0, (n-1)/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
A320221
Irregular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k, (n>=1, min(1,n-1) <= k <= log_2(n)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 7, 1, 1, 11, 4, 1, 13, 6, 1, 20, 16, 1, 23, 23, 1, 33, 46, 1, 40, 70, 1, 54, 127, 1, 1, 65, 189, 1, 1, 87, 320, 5, 1, 104, 476, 10, 1, 136, 771, 32, 1, 164, 1145, 63, 1, 209, 1795, 154, 1, 252, 2657, 304, 1, 319, 4091, 656
Offset: 1
Triangle begins:
1
1
1
1 1
1 1
1 3
1 3
1 6 1
1 7 1
1 11 4
1 13 6
1 20 16
1 23 23
1 33 46
1 40 70
The T(11,3) = 6 rooted trees:
(((oo)(oo))((oo)(ooooo)))
(((oo)(oo))((ooo)(oooo)))
(((oo)(ooo))((oo)(oooo)))
(((oo)(ooo))((ooo)(ooo)))
(((oo)(oo))((oo)(oo)(ooo)))
(((oo)(ooo))((oo)(oo)(oo)))
-
qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
DeleteCases[Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,10},{k,0,n-1}],0,{2}]
-
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); v[1]=x; [Vecrev(p/x) | p<-v]}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020
A334652
Number of integer partitions of n with at least two parts, each greater than 1 and with the same multiplicity.
Original entry on oeis.org
0, 0, 0, 0, 1, 1, 3, 2, 4, 5, 7, 6, 12, 9, 15, 17, 21, 20, 33, 28, 43, 44, 55, 55, 81, 77, 99, 108, 135, 136, 184, 180, 230, 246, 294, 316, 398, 403, 489, 532, 637, 668, 816, 852, 1019, 1107, 1275, 1370, 1637, 1727, 2016, 2185, 2518, 2701, 3152, 3370, 3884, 4200, 4774, 5154, 5953
Offset: 0
The a(4) = 1 partition is 2 + 2.
The a(7) = 2 partitions are 2 + 5 and 3 + 4. Each part has multiplicity 1.
-
Table[Length@Select[IntegerPartitions[n], Min[#] > 1 && Length[#] > 1 && (Length[Union[Length /@ Split[Sort[#]]]] == 1) &], {n, 0, 20}]
-
\\ here b(n) is A025147.
b(n)={my(A=O(x*x^n)); polcoef(eta(x^2 + A) / eta(x + A) / (1 + x), n)}
a(n)={if(n<=1, 0, sumdiv(n, d, b(d)) - 1)} \\ Andrew Howroyd, May 07 2020
A334653
Number of integer partitions of n with at least two parts, each greater than 1, at least two kinds of parts and all with the same multiplicity.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 8, 9, 13, 15, 18, 20, 29, 28, 39, 42, 53, 55, 75, 76, 97, 106, 131, 136, 178, 180, 226, 244, 292, 314, 391, 403, 487, 530, 631, 668, 810, 852, 1015, 1103, 1273, 1370, 1629, 1726, 2012, 2183, 2514, 2701, 3146, 3368, 3878, 4198
Offset: 0
The a(10) = 5 partitions are 2 + 8, 3 + 7, 4 + 6, 2 + 3 + 5 and 2 + 2 + 3 + 3.
-
Table[Length@Select[IntegerPartitions[n], Min[#] > 1 && Length[#] > 1 && Length[Union[#]] > 1 && (Length[Union[Length /@ Split[Sort[#]]]] == 1) &], {n, 0, 40}]
-
\\ here b(n) is A025147.
b(n)={my(A=O(x*x^n)); polcoef(eta(x^2 + A) / eta(x + A) / (1 + x), n)}
a(n)={if(n<1, 0, 1 + sumdiv(n, d, b(d)-1))} \\ Andrew Howroyd, May 07 2020
A320291
Number of singleton-free multiset partitions of integer partitions of n with no 1's.
Original entry on oeis.org
1, 0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 36, 46, 79, 110, 181, 254, 407, 580, 907, 1309, 2004, 2909, 4410, 6407, 9599, 13984, 20782, 30252, 44677, 64967, 95414, 138563, 202527, 293583, 427442, 618337, 897023, 1295020, 1872696, 2697777, 3889964, 5591917, 8041593, 11535890
Offset: 0
The a(4) = 1 through a(10) = 15 multiset partitions:
((22)) ((23)) ((24)) ((25)) ((26)) ((27)) ((28))
((33)) ((34)) ((35)) ((36)) ((37))
((222)) ((223)) ((44)) ((45)) ((46))
((224)) ((225)) ((55))
((233)) ((234)) ((226))
((2222)) ((333)) ((235))
((22)(22)) ((2223)) ((244))
((22)(23)) ((334))
((2224))
((2233))
((22222))
((22)(24))
((22)(33))
((23)(23))
((22)(222))
Cf.
A002865,
A007716,
A049311,
A083751,
A283877,
A293606,
A302545,
A304966,
A304967,
A320294,
A320295,
A320296.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]
-
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ Andrew Howroyd, Oct 25 2018
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