A063834 -id:A063834 - cp's OEIS frontend

A319794 Number of ways to split a strict integer partition of n into consecutive subsequences with weakly decreasing sums.

1, 1, 1, 3, 3, 5, 9, 11, 15, 20, 31, 37, 52, 64, 85, 111, 141, 175, 225, 279, 346, 437, 532, 654, 802, 979, 1182, 1438, 1740, 2083, 2502, 2996, 3565, 4245, 5043, 5950, 7068, 8303, 9772, 11449, 13452, 15681, 18355, 21338, 24855, 28846, 33509, 38687, 44819, 51644

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(6) = 9 split partitions:
    (6)
   (51)  (5)(1)
   (42)  (4)(2)
  (321)  (32)(1)  (3)(21)  (3)(2)(1).

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A318684.

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/@#,GreaterEqual]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A336130 Number of ways to split a strict composition of n into contiguous subsequences all having the same sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 15, 13, 23, 27, 73, 65, 129, 133, 241, 375, 519, 617, 1047, 1177, 1859, 2871, 3913, 4757, 7653, 8761, 13273, 16155, 28803, 30461, 50727, 55741, 87743, 100707, 152233, 168425, 308937, 315973, 500257, 571743, 871335, 958265, 1511583, 1621273, 2449259, 3095511, 4335385, 4957877, 7554717, 8407537, 12325993, 14301411, 20348691, 22896077, 33647199, 40267141, 56412983, 66090291, 93371665, 106615841, 155161833

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(7) = 13 splits:
  (1)  (2)  (3)    (4)    (5)    (6)        (7)
            (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)      (2,5)
                          (3,2)  (4,2)      (3,4)
                          (4,1)  (5,1)      (4,3)
                                 (1,2,3)    (5,2)
                                 (1,3,2)    (6,1)
                                 (2,1,3)    (1,2,4)
                                 (2,3,1)    (1,4,2)
                                 (3,1,2)    (2,1,4)
                                 (3,2,1)    (2,4,1)
                                 (1,2),(3)  (4,1,2)
                                 (2,1),(3)  (4,2,1)
                                 (3),(1,2)
                                 (3),(2,1)

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

The version with different instead of equal sums is A336128.
Starting with a non-strict composition gives A074854.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Set partitions with equal block-sums are A035470.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A317508, A318684, A326519, A336127, A336132, 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],SameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(60) from Max Alekseyev, Feb 14 2024

A358831 Number of twice-partitions of n into partitions with weakly decreasing lengths.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 56, 102, 205, 372, 708, 1260, 2345, 4100, 7388, 12819, 22603, 38658, 67108, 113465, 193876, 324980, 547640, 909044, 1516609, 2495023, 4118211, 6726997, 11002924, 17836022, 28948687, 46604803, 75074397, 120134298, 192188760, 305709858, 486140940

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(4) = 14 twice-partitions:
  (1)  (2)     (3)        (4)
       (11)    (21)       (22)
       (1)(1)  (111)      (31)
               (2)(1)     (211)
               (11)(1)    (1111)
               (1)(1)(1)  (2)(2)
                          (3)(1)
                          (11)(2)
                          (21)(1)
                          (11)(11)
                          (111)(1)
                          (2)(1)(1)
                          (11)(1)(1)
                          (1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

This is the semi-ordered case of A141199.
For constant instead of weakly decreasing lengths we have A306319.
For distinct instead of weakly decreasing lengths we have A358830.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A196545 counts p-trees, enriched A289501.
Cf. A000041, A000219, A001970, A061260, A072233, A271619, A279787, A358836.

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

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=Vec(P(n,y)-1), v=[1]); for(k=1, n, my(p=g[k], u=v); v=vector(k+1); v[1] = 1 + O(x*x^n); for(j=1, k, v[1+j] = (v[j] + if(jAndrew Howroyd, Dec 31 2022

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

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Original entry on oeis.org

1, 3, 5, 13, 17, 57, 65, 209, 321, 801, 1025, 3905, 4097, 12417, 21505, 53505, 65537, 233985, 262145, 885761, 1327105, 3147777, 4194305, 16060417, 17825793, 50339841, 84148225, 220217345, 268435457, 990937089, 1073741825, 3506503681

Offset: 1

Miklos Kristof, Sep 11 2002

A034729 = Sum_{d|n} (2^(d-1)).
A055895 = 2*A034729.
If p is a prime, then a(p) = A034729(p) = 2^(p-1)+1.
From Gus Wiseman, Jul 14 2020: (Start)
Number of ways to tile a rectangle of size n using horizontal strips. Also the number of ways to choose a composition of each part of a constant partition of n. The a(0) = 1 through a(5) = 17 splittings are:
  ()  (1)  (2)      (3)          (4)              (5)
           (1,1)    (1,2)        (1,3)            (1,4)
           (1),(1)  (2,1)        (2,2)            (2,3)
                    (1,1,1)      (3,1)            (3,2)
                    (1),(1),(1)  (1,1,2)          (4,1)
                                 (1,2,1)          (1,1,3)
                                 (2,1,1)          (1,2,2)
                                 (2),(2)          (1,3,1)
                                 (1,1,1,1)        (2,1,2)
                                 (1,1),(2)        (2,2,1)
                                 (2),(1,1)        (3,1,1)
                                 (1,1),(1,1)      (1,1,1,2)
                                 (1),(1),(1),(1)  (1,1,2,1)
                                                  (1,2,1,1)
                                                  (2,1,1,1)
                                                  (1,1,1,1,1)
                                                  (1),(1),(1),(1),(1)
(End)

			Divisors of 6 = 1,2,3,6 and 6-1 = 5, 6-2 = 4, 6-3 = 3, 6-6 = 0. a(6) = 2^5 + 2^4 + 2^3 + 2^0 = 32 + 16 + 8 + 1 = 57.
G.f. = x + 3*x^2 + 5*x^3 + 13*x^4 + 17*x^5 + 57*x^6 + 65*x^7 + ...
a(14) = 1 + 2^7 + 2^12 + 2^13 = 12417. - _Gus Wiseman_, Jun 20 2018

Gus Wiseman, Table of n, a(n) for n = 1..32

Cf. A055895, A034729.
Cf. A080267.
Cf. A051731.
The version looking at lengths instead of sums is A101509.
The strictly increasing (or strictly decreasing) version is A304961.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Requiring distinct instead of equal sums gives A336127.
Starting with a strict composition gives A336130.
Partitions of partitions are A001970.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Cf. A006951, A063834, A075900, A279787, A305551, A316245, A323433, A336128.

a[ n_] := If[ n < 1, 0, Sum[ 2^(n - d), {d, Divisors[n]}]] (* Michael Somos, Mar 28 2013 *)

a(n)=if(n<1,0,2^n*polcoeff(sum(k=1,n,2/(2-x^k),x*O(x^n)),n))

a(n) = sumdiv(n,d, 2^(n-d) ); /* Joerg Arndt, Mar 28 2013 */

G.f.: 2^n times coefficient of x^n in Sum_{k>=1} x^k/(2-x^k). - Benoit Cloitre, Apr 21 2003; corrected by Joerg Arndt, Mar 28 2013
G.f.: Sum_{k>0} 2^(k-1)*x^k/(1-2^(k-1)*x^k). - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=2*x and a=1/2. - Joerg Arndt, Jan 30 2011
Triangle A051731 mod 2 converted to decimal. - Philippe Deléham, Oct 04 2003
G.f.: Sum_{k>0} 1 / (2 / (2*x)^k - 1). - Michael Somos, Mar 28 2013

a(14) corrected from 9407 to 12417 by Gus Wiseman, Jun 20 2018

A279374 Number of ways to choose an odd partition of each part of an odd partition of 2n+1.

Original entry on oeis.org

1, 3, 6, 15, 37, 80, 183, 428, 893, 1944, 4223, 8691, 18128, 37529, 75738, 153460, 308829, 612006, 1211097, 2386016, 4648229, 9042678, 17528035, 33645928, 64508161, 123178953, 233709589, 442583046, 834923483, 1567271495, 2935406996, 5481361193, 10191781534

Offset: 0

Gus Wiseman, Dec 11 2016

nonn

An odd partition is an integer partition of an odd number with an odd number of parts, all of which are odd.

			The a(3)=15 ways to choose an odd partition of each part of an odd partition of 7 are:
((7)), ((511)), ((331)), ((31111)), ((1111111)), ((5)(1)(1)), ((311)(1)(1)),
((11111)(1)(1)), ((3)(3)(1)), ((3)(111)(1)), ((111)(3)(1)), ((111)(111)(1)),
((3)(1)(1)(1)(1)), ((111)(1)(1)(1)(1)), ((1)(1)(1)(1)(1)(1)(1)).

Alois P. Heinz, Table of n, a(n) for n = 0..4919
Gus Wiseman, "Twice-odd partitions of n=9."

Cf. A000009 (strict partitions), A078408 (odd partitions), A063834, A271619, A279375.

g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      [0, 2, 0, 1$4, 2, 0, 2, 1$4, 0, 2][1+irem(d, 16)],
      d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i, t) option remember;
      `if`(n=0, t, `if`(i<1, 0, b(n, i-2, t)+
      `if`(i>n, 0, b(n-i, i, 1-t)*g((i-1)/2))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 12 2016

nn=20;Table[SeriesCoefficient[Product[1/(1-PartitionsQ[k]x^k),{k,1,2n-1,2}],{x,0,2n-1}],{n,nn}]

A281119 Number of complete tree-factorizations of n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 9, 1, 1, 2, 3, 1, 4, 1, 12, 1, 1, 1, 12, 1, 1, 1, 9, 1, 4, 1, 3, 3, 1, 1, 29, 1, 3, 1, 3, 1, 9, 1, 9, 1, 1, 1, 17, 1, 1, 3, 34, 1, 4, 1, 3, 1, 4, 1, 44, 1, 1, 3, 3, 1, 4, 1, 29, 5, 1, 1

Offset: 2

Gus Wiseman, Jan 15 2017

nonn

A tree-factorization of n>=2 is either (case 1) the number n or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors greater than 1. A complete (or total) tree-factorization is a tree-factorization whose leaves are all prime numbers.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(36)=12 complete tree-factorizations of 36 are:
(2(2(33))), (2(3(23))), (2(233)),   (3(2(23))),
(3(3(22))), (3(223)),   ((22)(33)), ((23)(23)),
(22(33)),   (23(23)),   (33(22)),   (2233).

Michael De Vlieger, Table of n, a(n) for n = 2..10000
A. Knopfmacher, M. Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal 10(1), 2006.

Cf. A000311, A001055, A063834, A196545, A262673, A273873, A281113, A281118.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
treefacs[n_]:=If[n<=1,{{}},Prepend[Join@@Function[q,Tuples[treefacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[Select[treefacs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,83}]

seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=w[k]^e*v[i]))); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A196545(n) for prime p. - Andrew Howroyd, Nov 18 2018

A299925 Number of chains in Young's lattice from () to the partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 4, 12, 16, 16, 16, 32, 40, 44, 8, 64, 44, 128, 52, 136, 96, 256, 40, 88, 224, 88, 152, 512, 204, 1024, 16, 384, 512, 360, 136, 2048, 1152, 1024, 152, 4096, 744, 8192, 416, 496, 2560, 16384, 96, 720, 496, 2624, 1088, 32768, 360, 1216, 504

Offset: 1

Gus Wiseman, Feb 21 2018

nonn

a(n) is the number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions skew-partitions. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 12 tableaux:
1 3   1 2
2 4   3 4
.
1 3   1 2   1 2   1 2   1 1
2 3   3 3   2 3   1 3   2 3
.
1 2   1 2   1 1   1 1
2 2   1 2   2 2   1 2
.
1 1
1 1
The a(9) = 12 chains of Heinz numbers:
1<9,
1<2<9, 1<3<9, 1<4<9, 1<6<9,
1<2<3<9, 1<2<4<9, 1<2<6<9, 1<3<6<9, 1<4<6<9,
1<2<3<6<9, 1<2<4<6<9.

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A296561, A297388, A299202.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hncQ[a_,b_]:=And@@GreaterEqual@@@Transpose[PadRight[{Reverse[primeMS[b]],Reverse[primeMS[a]]}]];
chns[x_,y_]:=chns[x,y]=Join[{{x,y}},Join@@Function[c,Append[#,y]&/@chns[x,c]]/@Select[Range[x+1,y-1],hncQ[x,#]&&hncQ[#,y]&]];
Table[Length[chns[1,n]],{n,30}]

A299926 a(n) is the number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions skew partitions.

Original entry on oeis.org

1, 4, 14, 60, 252, 1212, 5880, 30904, 166976, 952456, 5587840, 34217216, 215204960, 1401551376, 9360467760, 64384034784, 453328282624, 3274696185568, 24173219998912, 182546586425408

Offset: 1

Gus Wiseman, Feb 21 2018

If y is an integer partition of n, a generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(3) = 14 tableaux:
1 2 3   1 2 2   1 1 2   1 1 1
.
1 3   1 2   1 2   1 2   1 1   1 1
2     3     2     1     2     1
.
1   1   1   1
2   2   1   1
3   2   2   1

Cf. A000085, A063834, A138178, A153452, A296188, A296561, A297388, A299699, A299925.

undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
chn[y_]:=Join[{{{},y}},Join@@Function[c,Append[#,y]&/@chn[c]]/@Take[undptns[y],{2,-2}]];
Table[Sum[Length[chn[y]],{y,IntegerPartitions[n]}],{n,8}]

A301706 Number of rooted thrice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 91, 201, 422, 918, 1896, 4089, 8376, 17793, 36445, 76446, 155209, 324481, 655426, 1355220, 2741092, 5617505, 11291037, 23086423, 46227338, 93753196, 187754647, 378675055, 754695631, 1518414812, 3016719277, 6037006608, 11984729983

Offset: 1

Gus Wiseman, Mar 25 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. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

			The a(5) = 9 rooted thrice-partitions:
((2)), ((11)), ((1)()), (()()()),
((1))(), (()())(), (())(()),
(())()(),
()()()().
The a(6) = 19 rooted thrice-partitions:
((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
((1))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.

twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1),{ptn,IntegerPartitions[n-1]}];
thrire[n_]:=Sum[Times@@twire/@ptn,{ptn,IntegerPartitions[n-1]}];
Array[thrire,30]

A318683 Number of ways to split a strict integer partition of n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 5, 7, 8, 12, 12, 18, 18, 26, 27, 37, 38, 53, 54, 73, 76, 100, 104, 136, 142, 183, 192, 244, 256, 327, 340, 424, 448, 558, 585, 722, 760, 937, 983, 1195, 1260, 1544, 1610, 1943, 2053, 2480, 2590, 3107, 3264, 3927, 4106, 4874, 5120, 6134, 6378

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(12) = 18 constant-sum split partitions:
  (12)
  (7,5)
  (8,4)
  (9,3)
  (10,2)
  (11,1)
  (5,4,3)
  (6,4,2)
  (6,5,1)
  (7,3,2)
  (7,4,1)
  (8,3,1)
  (9,2,1)
  (6)(4,2)
  (6)(5,1)
  (5,4,2,1)
  (6,3,2,1)
  (6)(3,2,1)

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A318684, 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],SameQ@@Total/@#&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

cp's OEIS Frontend

A319794 Number of ways to split a strict integer partition of n into consecutive subsequences with weakly decreasing sums.

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A336130 Number of ways to split a strict composition of n into contiguous subsequences all having the same sum.

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Mathematica

Extensions

A358831 Number of twice-partitions of n into partitions with weakly decreasing lengths.

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A074854 a(n) = Sum_{d|n} (2^(n-d)).

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A279374 Number of ways to choose an odd partition of each part of an odd partition of 2n+1.

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A281119 Number of complete tree-factorizations of n >= 2.

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A299925 Number of chains in Young's lattice from () to the partition with Heinz number n.

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Mathematica

A299926 a(n) is the number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions skew partitions.

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