A292432 -id:A292432 - cp's OEIS frontend

A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.

1, 1, 3, 4, 7, 13, 21, 34, 58, 98, 158, 258, 421, 676, 1108, 1777, 2836, 4544, 7220, 11443, 18215, 28729, 45203, 71139, 111518, 174402, 272367, 424892, 660563, 1025717, 1590448, 2460346, 3800816, 5862640, 9026963, 13885425, 21321663, 32695098, 50073855

Offset: 1

Gus Wiseman, Jan 31 2018

nonn

			The a(5) = 7 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (212) = (2)*(12)
Not included:
    (311) = (3)*(1)*(1)
    (113) = (113)
    (221) = (2)*(2)*(1)
    (122) = (122)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
   (1121) = (112)*(1)
   (1112) = (1112)
  (11111) = (1)*(1)*(1)*(1)*(1)

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

Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877, A292432, A296373, A299023, A299026, A299027.

nn=50;
ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(N)={WeighT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018

Weigh transform of A032153.

A299027 Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.

Original entry on oeis.org

1, 1, 3, 5, 11, 20, 38, 69, 125, 225, 400, 708, 1244, 2176, 3779, 6532, 11229, 19223, 32745, 55555, 93875, 158025, 265038, 443009, 738026, 1225649, 2029305, 3350167, 5515384, 9055678, 14830076, 24226115, 39480306, 64190026, 104130753, 168556588, 272268482

Offset: 1

Gus Wiseman, Feb 01 2018

nonn

			The a(5) = 11 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (113) = (113)
    (212) = (2)*(12)
    (122) = (122)
   (1121) = (112)*(1)
   (1112) = (1112)
Not included:
    (311) = (3)*(1)*(1)
    (221) = (2)*(2)*(1)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
  (11111) = (1)*(1)*(1)*(1)*(1)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A116540, A185700, A270995, A283877, A292432, A293993, A296373, A299023, A299024, A299026.

nn=50;
ser=Product[(1+x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={WeighT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

Weigh transform of A167934.

A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 0, 0, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(180) = 4 strict set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2},{3},{1,2}}

Cf. A007716, A045778, A049311, A050326, A056239, A112798, A116539, A116540, A292432, A292444, A293511, A318284, A318360, A318369.

A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

Original entry on oeis.org

1, 2, 6, 20, 74, 311, 1401, 6913, 36376, 205421, 1228288, 7786802, 51937607, 364250763, 2673314121, 20504809133, 163844631872, 1361874185139, 11748149246269, 105029750531640, 971403871953460, 9282643841237360, 91519776792040324, 929892817423282068, 9725646244888190337

Offset: 1

Gus Wiseman, Aug 25 2018

nonn

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(4) = 20 sets of sets:
  {{1,2,3,4}}
  {{1},{1,2,3}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{3,4}}
  {{1},{3},{1,2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A007716, A049311, A050326, A116540, A283877, A292432, A292444, A318361, A318369, A318370.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

cp's OEIS Frontend

A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.

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A299027 Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.

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A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

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A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

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