A319182 -id:A319182 - cp's OEIS frontend

A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 1, 5, 10, 1, 1, 15, 1, 10, 15, 6, 1, 10, 10, 7, 15, 15, 1, 60, 1, 1, 21, 8, 35, 45, 1, 9, 28, 20, 1, 105, 1, 21, 105, 10, 1, 15, 35, 70, 36, 28, 1, 105, 56, 35, 45, 11, 1, 210, 1, 12, 210, 1, 84, 168, 1, 36, 55, 280, 1, 105, 1, 13, 280, 45, 126, 252, 1

Offset: 1

Max Alekseyev, Nov 07 2006

nonn

Coefficients of (D^k f)(g(t))*(D g(t))^k1*(D^2 g(t))^k2*... in the Faa di Bruno formula for D^m(f(g(t))) where k = k1 + k2 + ..., m = 1*k1 + 2*k2 + ....

Number of set partitions whose block sizes are the prime indices of n (i.e., the integer partition with Heinz number n). - Gus Wiseman, Sep 12 2018

			The a(6) = 3 set partitions of type (2,1) are {{1},{2,3}}, {{1,3},{2}}, {{1,2},{3}}. - _Gus Wiseman_, Sep 12 2018

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Faà di Bruno's Formula

Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.

Cf. A065475, A100484.

with(numtheory):
a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i],
         i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)),
         i=1..pi(max(1, factorset(n))))]):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 14 2020

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[numSetPtnsOfType[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}] (* Gus Wiseman, Sep 12 2018 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k,1])*f[k,2])!/(prod(k=1, #f~, f[k,2]!)*prod(k=1, #f~, primepi(f[k,1])!^f[k,2])); \\ Michel Marcus, Oct 11 2023

For n = p1^k1*p2^k2*... where 2 = p1 < p2 < ... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1!^k1*2!^k2*...)).

a(2*prime(n)) = n + 1, for n > 1. See A065475. - Bill McEachen, Oct 11 2023

A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 1, 3, 6, 6, 4, 5, 1, 1, 2, 2, 2, 6, 3, 3, 3, 4, 4, 12, 10, 5, 6, 1, 1, 2, 2, 1, 3, 2, 3, 6, 6, 3, 1, 12, 4, 12, 6, 10, 5, 20, 15, 6, 7, 1, 1, 2, 2, 2, 3, 2, 6, 3, 3, 4, 6, 6, 1, 12, 12, 4, 12

Offset: 0

Gus Wiseman, Sep 13 2018

A refinement of Pascal's triangle, these are the unsigned coefficients appearing in the expansion of homogeneous symmetric functions in terms of elementary symmetric functions.

			Triangle begins:
  1
  1
  1  1
  1  2  1
  1  1  2  3  1
  1  2  2  3  3  4  1
  1  2  2  1  1  3  6  6  4  5  1
The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).

Alois P. Heinz, Rows n = 0..33, flattened

A different row ordering is A072811.

Cf. A000041, A005651, A008277, A008480, A056239, A124794, A215366, A319182, A319191, A319192.

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(
      map(p-> p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> map(m-> (l-> add(i, i=l)!/mul(i!, i=l))(map(
        i-> i[2], ifactors(m)[2])), sort(b(n$2)))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Permutations[primeMS[k]]],{n,6},{k,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0 || i < 2, {2^n}, Flatten[Table[ #*Prime[i]^j& /@ b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Map[Function[m, Function[l, Total[l]!/Times @@ (l!)][ FactorInteger[m][[All, 2]]]], Sort[b[n, n]]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

T(n,k) = A008480(A215366(n,k)).

T(0,1)=1 prepended by Alois P. Heinz, Feb 14 2020

A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 1, 2, -3, -6, 1, 3, 8, 24, -6, -120, -30, -20, 1, 720, 15, -5040, 20, 90, 144, 40320, -10, 40, -840, -15, -90, -362880, -120, 3628800, 1, -504, 5760, -420, 45, -39916800, -45360, 3360, 40, 479001600, 630, -6227020800, 504, 210, 403200, 87178291200

Offset: 1

Gus Wiseman, Sep 13 2018

sign

A refinement of Stirling numbers of the first kind.

An unsigned version is A124795.

Cf. A000041, A000110, A000258, A005651, A008480, A048994, A056239, A124794, A215366, A318762, A319182.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[n]]-PrimeOmega[n])*numPermsOfType[primeMS[n]],{n,100}]

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (-1)^(Sum x_i * y_i - Sum y_i) (Sum x_i * y_i)! / (Product x_i^y_i * Product y_i!).

A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 3, 6, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
   1
   1
   1   0
   1   2
   1   0   0
   1   1   0
   1   0   0   0   0
   1   3   6
   1   2   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
   1   2   2   2   0
   1   0   0   0   0   0   0   0   0   0   0
   1   1   0   0   0   0   0
   1   0   1   0   0   0   0
   1   6   4  12  24
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   1   1   2   2   0   0   0
For example, row 18 gives: p(221) = m(5) + 2m(32) + m(41) + 2m(221).

Wikipedia, Symmetric polynomial

Row sums are A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A321742-A321765.

A321751 Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 10, 3, 2, 1, 7, 1, 2, 2, 47, 1, 6, 1, 6, 2, 2, 1, 26, 3, 2, 10, 6, 1, 6, 1, 246, 2, 2, 2, 26, 1, 2, 2, 24, 1, 5, 1, 6, 6, 2, 1, 138, 3, 6, 2, 6, 1, 23, 2, 23, 2, 2, 1, 20, 1, 2, 7, 1602, 2, 5, 1, 6, 2, 6, 1, 105, 1, 2, 6, 6, 2, 5, 1, 114

Offset: 1

Gus Wiseman, Nov 20 2018

nonn

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

Also the number of ordered set partitions of {1, 2, ..., A001222(n)} whose blocks, when i is replaced by the i-th prime index of n, have weakly decreasing sums.

			The sum of coefficients of p(211) = m(4) + 2m(22) + 2m(31) + 2m(211) is a(12) = 7.

Wikipedia, Symmetric polynomial

Row sums of A321750.

Cf. A005651, A008277, A008480, A056239, A124794, A124795, A296150, A319182, A319225, A319226, A321742-A321765.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Total/@s]],{s,sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[i]],{i,PrimeOmega[n]}]}],{n,50}]

A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -6, 3, 8, -6, 1, 24, -30, -20, 15, 20, -10, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 720, -840, -504, -420, 630, 504, 210, 280, -105, -210, -420, 105, 70, -21, 1, -5040, 5760, 3360, 1260, -3360, 2688, -1260, -4032, -3360, -1120

Offset: 1

Gus Wiseman, Sep 13 2018

A generalization of the triangle of Stirling numbers of the first kind, these are the coefficients appearing in the expansion of single-part augmented elementary symmetric functions in terms of power-sum symmetric functions.

			Triangle begins:
   1
  -1   1
   2  -3   1
  -6   3   8  -6   1
  24 -30 -20  15  20 -10   1
The fourth row corresponds to the symmetric function identity: 24 e(4) = -6 p(4) + 3 p(22) + 8 p(31) - 6 p(211) + p(1111).

Other row orderings are A036039 and A102189.

Cf. A000041, A005651, A008277, A008480, A048994, A056239, A124794, A124795, A215366, A318762, A319182, A319191.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*numPermsOfType[primeMS[m]],{n,5},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

A321888 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and f is forgotten symmetric functions.

Original entry on oeis.org

1, 1, -1, 0, 1, 2, 1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 1, 3, 6, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -2, -2, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Up to sign, a(n) is also the coefficient of m(v) in p(u), where m is monomial symmetric functions.

			Triangle begins:
   1
   1
  -1   0
   1   2
   1   0   0
  -1  -1   0
  -1   0   0   0   0
   1   3   6
   1   2   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
  -1  -2  -2  -2   0
  -1   0   0   0   0   0   0   0   0   0   0
  -1  -1   0   0   0   0   0
  -1   0  -1   0   0   0   0
   1   6   4  12  24
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   1   1   2   2   0   0   0
For example, row 12 gives: p(211) = -f(4) - 2f(22) - 2f(31) - 2f(211).

Wikipedia, Symmetric polynomial

Row sums are A321889.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319193, A321742-A321765.

A318120 Number of set partitions of {1,...,n} with relatively prime block sizes.

Original entry on oeis.org

1, 1, 1, 4, 11, 51, 162, 876, 3761, 20782, 109293, 678569, 4038388, 27644436, 186524145, 1379760895, 10323844183, 82864869803, 674798169662, 5832742205056, 51385856585637, 474708148273586, 4486977535287371, 44152005855084345, 444577220573083896

Offset: 0

Gus Wiseman, Dec 16 2018

nonn

			The a(4) = 11 set partitions:
  {{1},{2},{3},{4}}
   {{1},{2},{3,4}}
   {{1},{2,3},{4}}
   {{1},{2,4},{3}}
   {{1,2},{3},{4}}
   {{1,3},{2},{4}}
   {{1,4},{2},{3}}
    {{1},{2,3,4}}
    {{1,2,3},{4}}
    {{1,2,4},{3}}
    {{1,3,4},{2}}

Alois P. Heinz, Table of n, a(n) for n = 0..576

a(n) >= A271426(n).

Cf. A000110, A000258, A000311, A000670, A000740, A000837, A005651, A005804, A008277, A124794, A319182, A320289.

b:= proc(n, t) option remember; `if`(n=0, `if`(t<2, 1, 0),
      add(b(n-j, igcd(t, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2019

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[Total[numSetPtnsOfType/@Select[IntegerPartitions[n],GCD@@#==1&]],{n,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, If[t < 2, 1, 0],
     Sum[b[n - j, GCD[t, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = Sum_{|y| = n, GCD(y) = 1} n! / (Product_i y_i! * Product_i (y)_i!) where the sum is over all relatively prime integer partitions of n and (y)_i is the multiplicity of i in y.

A321887 Sum of coefficients of monomial symmetric functions in the forgotten symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 2, 1, -3, -1, 3, 2, 3, 1, -8, -1, -3, -3, 5, 1, 7, -1, 7, 3, 3, 1, -15

Offset: 1

Gus Wiseman, Nov 20 2018

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

			The sum of coefficients of m(221) = 3m(5) + 2m(32) + m(41) + m(221) is a(18) = 7.

Wikipedia, Symmetric polynomial

Row sums of A321886.

Cf. A005651, A008277, A008480, A056239, A124794, A124795, A296150, A319182, A319193, A321742-A321765.

A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 3, 1, -2, -1, 10, 3, 2, 1, -7, -1, -2, -2, 47, 1, 6

Offset: 1

Gus Wiseman, Nov 20 2018

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

			The sum of coefficients of p(211) = -f(4) - 2f(22) - 2f(31) - 2f(211) is a(12) = -7.

Wikipedia, Symmetric polynomial

Row sums of A321888. An unsigned version is A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A319193, A321742-A321765.

cp's OEIS Frontend

A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

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A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).

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A319191 Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.

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A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

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A321751 Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

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Programs

Mathematica

A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

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A321888 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and f is forgotten symmetric functions.

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A318120 Number of set partitions of {1,...,n} with relatively prime block sizes.

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