A063834 -id:A063834 - cp's OEIS frontend

A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)prime_m(2)...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).

1, 2, 3, 3, 4, 5, 5, 4, 6, 7, 6, 7, 7, 9, 9, 5, 8, 9, 9, 10, 12, 11, 10, 9, 10, 13, 10, 13, 11, 14, 12, 6, 15, 15, 14, 12, 13, 17, 18, 13, 14, 19, 15, 16, 16, 19, 16, 11, 15, 16, 21, 19, 17, 14, 18, 17, 24, 21, 18, 19, 19, 23, 22, 7, 22, 24, 20, 22, 27, 23, 21

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Equivalently, a(n) equals the number of values of m such that each value of A238689 T(m,k) <= A238689 T(n,k). (Since the prime factorization of 1 is the empty factorization, we consider each prime_1(i) not to be greater than prime_n(i) for all positive integers n.)
Suppose we say that n "covers" m iff both m and n are factorized as described in the sequence definition and each prime_m(i) <= prime_n(i). At least three sequences (A037019, A108951 and A181821) have the property that a(m) divides a(n) iff n "covers" m.  These sequences are also divisibility sequences (i.e., sequences with the property that a(m) divides a(n) if m divides n), since any positive integer "covers" each of its divisors.
For any positive integers m and k, the following integer sequences (with n >= 0) are arithmetic progressions:
1. The sequence b(n) = a(m*(2^n)).
2. The sequence b(n) = a(m*(prime(n+k))) if prime(k) >= A006530(m).
Also, a(n) = the number of distinct prime signatures that occur among the divisors of any integer m such that A181819(m) = n and/or A238745(m) = n.
Number of skew partitions whose numerator has Heinz number n, where a skew partition is a pair y/v of integer partitions such that the diagram of v fits inside the diagram of y. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Feb 24 2018

			The prime factorizations of integers 1 through 9, with prime factors sorted from largest to smallest:
1 - the empty factorization (no prime factors)
2 = 2
3 = 3
4 = 2*2
5 = 5
6 = 3*2
7 = 7
8 = 2*2*2
9 = 3*3
To find a(9), we consider 9 = 3*3. There are 6 positive integers (1, 2, 3, 4, 6 and 9) which satisfy the following criteria:
1) The largest prime factor, if one exists, is not greater than 3;
2) The second-largest prime factor, if one exists, is not greater than 3;
3) The total number of prime factors (counting repeated factors) does not exceed 2.
Therefore, a(9) = 6.
From _Gus Wiseman_, Feb 24 2018: (Start)
Heinz numbers of the a(15) = 9 partitions contained within the partition (32) are 1, 2, 3, 4, 5, 6, 9, 10, 15. The a(15) = 9 skew partitions are (32)/(), (32)/(1), (32)/(11), (32)/(2), (32)/(21), (32)/(22), (32)/(3), (32)/(31), (32)/(32).
Corresponding diagrams are:
  o o o   . o o   . o o   . . o   . . o   . . o   . . .   . . .   . . .
  o o     o o     . o     o o     . o     . .     o o     . o     . .    (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Rearrangement of A115728, A115729 and A238746. A116473(n) is the number of times n appears in the sequence.

Cf. A000041, A000085, A000720, A056239, A063834, A112798, A122111, A153452, A215366, A238689, A259478, A259480, A296150, A296188, A296561, A297388, A299925, A299926, A299966, A299967.

undptns[y_]:=Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[undptns[Reverse[primeMS[n]]]],{n,100}] (* Gus Wiseman, Feb 24 2018 *)

a(n) = A085082(A108951(n)) = A085082(A181821(n)).
a(n) = a(A122111(n)).
a(prime(n)) = a(2^n) = n+1.
a((prime(n))^m) = a((prime(m))^n) = binomial(n+m, n).
a(A002110(n)) = A000108(n+1).
A000005(n) <= a(n) <= n.

A296561 Number of rim-hook (or border-strip) tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 4, 10, 12, 16, 12, 32, 28, 29, 8, 64, 29, 128, 33, 78, 64, 256, 28, 62, 144, 62, 86, 512, 100, 1024, 16, 200, 320, 193, 78, 2048, 704, 496, 86, 4096, 306, 8192, 216, 242, 1536, 16384, 64, 414, 242, 1200, 528, 32768, 193, 552, 245, 2848, 3328

Offset: 1

Gus Wiseman, Feb 15 2018

nonn

The Murnaghan-Nakayama rule uses rim-hook tableaux to expand Schur functions in terms of power-sum symmetric functions.

			The a(6) = 5 tableaux:
3 2   3 1   2 2   2 1   1 1
1     2     1     2     1

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.17.

Wikipedia, Murnaghan-Nakayama rule

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A296150, A296188, A299202.

A299203 Number of enriched p-trees whose multiset of leaves is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 5, 1, 12, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 38, 1, 3, 1, 3, 1, 9, 1, 9, 1, 1, 1, 21, 1, 1, 4, 34, 1, 4, 1, 3, 1, 5, 1, 54, 1, 1, 3, 3, 1, 4, 1, 33, 5, 1, 1, 23, 1, 1, 1, 9, 1, 20, 1, 3, 1, 1, 1, 117, 1, 3, 3, 12, 1, 4, 1, 9, 4, 1, 1, 57, 1, 4, 1, 34

Offset: 1

Gus Wiseman, Feb 05 2018

nonn

By convention, a(1) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			a(54) = 9: (((22)2)1), ((222)1), (((22)1)2), (((21)2)2), ((221)2), ((22)(21)), ((22)21), ((21)22), (2221).
a(40) = 11: ((31)(11)), (((31)1)1), ((3(11))1), ((311)1), (3((11)1)), (3(111)), (((11)1)3), ((111)3), ((31)11), (3(11)1), (3111).
a(36) = 15: ((22)(11)), ((2(11))2), (((11)2)2), (((21)1)2), ((211)2), (((22)1)1), (((21)2)1), ((221)1), ((21)(21)), (22(11)), (2(11)2), ((11)22), ((22)11), ((21)21), (2211).

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299202.

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
qci[y_]:=qci[y]=If[Length[y]===1,1,Sum[Times@@qci/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y]&]}]];
qci/@ptns

A301480 Number of rooted twice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 54, 103, 186, 345, 606, 1115, 1936, 3466, 6046, 10630, 18257, 31927, 54393, 93894, 159631, 272155, 458891, 779375, 1305801, 2196009, 3667242, 6130066, 10170745, 16923127, 27942148, 46211977, 76039205, 125094369, 204952168, 335924597

Offset: 1

Gus Wiseman, Mar 22 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(5) = 8 rooted twice-partitions: ((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()).
The a(6) = 15 rooted twice-partitions:
(4), (31), (22), (211), (1111),
(3)(), (21)(), (111)(), (2)(1), (11)(1),
(2)()(), (11)()(), (1)(1)(),
(1)()()(),
()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001383, A002865, A032305, A063834, A093637, A127524, A196545, A220418, A281113, A289501, A301422, A301462, A301467.

nn=30;
ser=x*Product[1/(1-PartitionsP[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(1/prod(k=1, n-1, 1 - numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} 1/(1 - P(n-1) x^n) where P = A000041.

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 6, 2, 1, 7, 11, 6, 2, 1, 11, 23, 15, 6, 2, 1, 15, 40, 32, 15, 6, 2, 1, 22, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 6, 2, 1, 56, 320, 470, 348, 187, 85, 37, 15, 6, 2, 1, 77, 511, 848, 697, 397, 194, 85, 37, 15, 6, 2, 1

Offset: 1

Vladeta Jovovic, Apr 23 2001

Multiset transformation of A000041. - R. J. Mathar, Apr 30 2017

Number of orderless twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. The T(5,3) = 6 orderless twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(11)(1). - Gus Wiseman, Mar 23 2018

			:  1;
:  2,   1;
:  3,   2,   1;
:  5,   6,   2,   1;
:  7,  11,   6,   2,  1;
: 11,  23,  15,   6,  2,  1;
: 15,  40,  32,  15,  6,  2,  1;
: 22,  73,  67,  37, 15,  6,  2, 1;
: 30, 120, 134,  79, 37, 15,  6, 2, 1;
: 42, 202, 255, 172, 85, 37, 15, 6, 2, 1;

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation

Row sums: A001970, first column: A000041.
T(2,n) gives A061261,
Cf. A063834, A119442, A273873, A285229, A289078, A289501, A299200, A299201.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(
       combinat[numbpart](i)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[PartitionsP[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 17 2018, after Alois P. Heinz *)

A300301 Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 15, 21, 37, 56, 80, 127, 183, 280, 428, 616, 893, 1367, 1944, 2846, 4223, 6049, 8691, 12670, 18128, 25921, 37529, 53338, 75738, 108561, 153460, 216762, 308829, 433893, 612006, 864990, 1211097, 1697020, 2386016, 3331037, 4648229, 6503314

Offset: 0

Gus Wiseman, Mar 02 2018

nonn

			The a(6) = 10 twice-partitions using odd partitions: (5)(1), (3)(3), (113)(1), (3)(111), (111)(3), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

A300300(n) <= a(n) <= A279785(n)

Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279790, A294617, A300300.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-2)+`if`(i>n, 0, b(i)*g(n-i, i)))
    end:
a:= n-> g(n, n-1+irem(n,2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 05 2018

nn=50;
ser=Product[1/(1-PartitionsQ[n]x^n),{n,1,nn,2}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

O.g.f.: Product_{n odd} 1/(1 - A000009(n)x^n).

A318684 Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 11, 14, 20, 28, 35, 48, 61, 79, 105, 129, 162, 208, 257, 318, 404, 489, 600, 732, 896, 1075, 1315, 1576, 1895, 2272, 2715, 3217, 3851, 4537, 5377, 6353, 7484, 8765, 10314, 12044, 14079, 16420, 19114, 22184, 25818, 29840, 34528, 39903, 46030

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(9) = 20 split partitions:
    (9)
   (81)   (8)(1)
   (72)   (7)(2)
   (63)   (6)(3)
   (54)   (5)(4)
  (432)  (43)(2)  (4)(3)(2)
  (621)  (62)(1)  (6)(2)(1)  (6)(21)
  (531)  (53)(1)  (5)(3)(1)  (5)(31)

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],OrderedQ[Total/@#,Greater]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A358830 Number of twice-partitions of n into partitions with all different lengths.

Original entry on oeis.org

1, 1, 2, 4, 9, 15, 31, 53, 105, 178, 330, 555, 1024, 1693, 2991, 5014, 8651, 14242, 24477, 39864, 67078, 109499, 181311, 292764, 483775, 774414, 1260016, 2016427, 3254327, 5162407, 8285796, 13074804, 20812682, 32733603, 51717463, 80904644, 127305773, 198134675, 309677802

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)      (4)       (5)
       (11)  (21)     (22)      (32)
             (111)    (31)      (41)
             (11)(1)  (211)     (221)
                      (1111)    (311)
                      (11)(2)   (2111)
                      (2)(11)   (11111)
                      (21)(1)   (21)(2)
                      (111)(1)  (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)

The version for set partitions is A007837.
For sums instead of lengths we have A271619.
For constant instead of distinct lengths we have A306319.
The case of distinct sums also is A358832.
The version for multiset partitions of integer partitions is A358836.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A273873 counts strict trees.
Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A336342, A358334, A358831.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Length/@#&]],{n,0,10}]

seq(n)={ local(Cache=Map());
  my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
  my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
  z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m],k)); if(!bittest(b,k)&&c, c*self()(min(m,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A300486 Number of relatively prime or monic partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 15, 18, 28, 35, 56, 64, 101, 120, 168, 210, 297, 348, 490, 583, 776, 946, 1255, 1482, 1952, 2335, 2981, 3581, 4565, 5387, 6842, 8119, 10086, 12013, 14863, 17527, 21637, 25525, 31083, 36695, 44583, 52256, 63261, 74171, 88932, 104303, 124754

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

			The a(6) = 8 relatively prime or monic partitions are (6), (51), (411), (321), (3111), (2211), (21111), (111111). Missing from this list are (42), (33), (222).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A301462, A301467, A301480, A302094, A302698, A302915, A302916, A302917.

Table[Length[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]],{n,20}]

a(n)={(n > 1) + sumdiv(n, d, moebius(d)*numbpart(n/d))} \\ Andrew Howroyd, Aug 29 2018

a(n > 1) = 1 + A000837(n) = 1 + Sum_{d|n} mu(d) * A000041(n/d).

cp's OEIS Frontend

A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)*prime_m(2)*...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).

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A296561 Number of rim-hook (or border-strip) tableaux whose shape is the integer partition with Heinz number n.

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A299203 Number of enriched p-trees whose multiset of leaves is the integer partition with Heinz number n.

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A301480 Number of rooted twice-partitions of n.

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Mathematica

PARI

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A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

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Mathematica

Extensions

A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.

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A300301 Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.

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Maple

Mathematica

Formula

A318684 Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.

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A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)prime_m(2)...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).