A124794 -id:A124794 - cp's OEIS frontend

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

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

A321742 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 0, 1, 1, 3, 6, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 19 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
   0   1
   1   2
   0   0   1
   0   1   3
   0   0   0   0   1
   1   3   6
   0   1   0   2   6
   0   0   0   1   4
   0   0   0   0   0   0   1
   0   2   1   5  12
   0   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   1   5
   0   0   0   1   0   3  10
   1   6   4  12  24
   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
   0   0   1   5   2  12  30
For example, row 12 gives: e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111).

Wikipedia, Symmetric polynomial

Row sums are A321743.

Cf. A008480, A049311, A056239, A116540, A124794, A124795, A300121, A319193, A321738, A321742-A321765, A321854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],And[And@@UnsameQ@@@#,Sort[Length/@#]==primeMS[k]]&]}],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}],{n,18}]

A134264 Coefficients T(j, k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions and primitive parking functions. T(n,k) for n >= 1 and 1 <= k <= A000041(n-1), an irregular triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 2, 6, 1, 1, 5, 5, 10, 10, 10, 1, 1, 6, 6, 3, 15, 30, 5, 20, 30, 15, 1, 1, 7, 7, 7, 21, 42, 21, 21, 35, 105, 35, 35, 70, 21, 1, 1, 8, 8, 8, 4, 28, 56, 56, 28, 28, 56, 168, 84, 168, 14, 70, 280, 140, 56, 140, 28, 1, 1, 9, 9, 9, 9, 36, 72

Offset: 1

Tom Copeland, Jan 14 2008

Coefficients are listed in Abramowitz and Stegun order (A036036).
Given an invertible function f(t) analytic about t=0 (or a formal power series) with f(0)=0 and Df(0) not equal to 0, form h(t) = t / f(t) and let h_n denote the coefficient of t^n in h(t).
Lagrange inversion gives the compositional inverse about t=0 as g(t) = Sum_{j>=1} ( t^j * (1/j) * Sum_{permutations s with s(1) + s(2) + ... + s(j) = j - 1} h_s(1) * h_s(2) * ... * h_s(j) ) = t * T(1,1) * h_0 + Sum_{j>=2} ( t^j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) ), where H(j-1,k ; h_0,h_1,...) is the k-th partition for h_1 through h_(j-1) corresponding to n=j-1 on page 831 of Abramowitz and Stegun (ordered as in A&S) with (h_0)^(j-m)=(h_0)^(n+1-m) appended to each partition subsumed under n and m of A&S.
Denoting h_n by (n') for brevity, to 8th order in t,
g(t) = t * (0')
+ t^2 * [ (0') (1') ]
+ t^3 * [ (0')^2 (2') + (0') (1')^2 ]
+ t^4 * [ (0')^3 (3') + 3 (0')^2 (1') (2') + (0') (1')^3 ]
+ t^5 * [ (0')^4 (4') + 4 (0')^3 (1') (3') + 2 (0')^3 (2')^2 + 6 (0')^2 (1')^2 (2') + (0') (1')^4 ]
+ t^6 * [ (0')^5 (5') + 5 (0')^4 (1') (4') + 5 (0')^4 (2') (3') + 10 (0')^3 (1')^2 (3') + 10 (0')^3 (1') (2')^2 + 10 (0')^2 (1')^3 (2') + (0') (1')^5 ]
+ t^7 * [ (0')^6 (6') + 6 (0')^5 (1') (5') + 6 (0')^5 (2') (4') + 3 (0')^5 (3')^2 + 15 (0')^4 (1')^2 (4') + 30 (0')^4 (1') (2') (3') + 5 (0')^4 (2')^3 + 20 (0')^3 (1')^3 (3') + 30 (0')^3 (1')^2 (2')^2 + 15 (0')^2 (1')^4 (2') + (0') (1')^6]
+ t^8 * [ (0')^7 (7') + 7 (0')^6 (1') (6') + 7 (0')^6 (2') (5') + 7  (0')^6 (3') (4') + 21 (0')^5 (1')^2* (5') + 42 (0')^5 (1') (2') (4') + 21 (0')^5 (1') (3')^2 + 21 (0')^5 (2')^2 (3') + 35 (0')^4 (1')^3 (4') + 105 (0)^4 (1')^2 (2') (3') + 35 (0')^4 (1') (2')^3  + 35 (0')^3 (1')^4 (3') + 70 (0')^3 (1')^3 (2')^2 + 21 (0')^2 (1')^5 (2') + (0') (1')^7 ]
+ ..., where from the formula section, for example, T(8,1',2',...,7') = 7! / ((8 - (1'+ 2' + ... + 7'))! * 1'! * 2'! * ... * 7'!) are the coefficients of the integer partitions (1')^1' (2')^2' ... (7')^7' in the t^8 term.
A125181 is an extended, reordered version of the above sequence, omitting the leading 1, with alternate interpretations.
If the coefficients of partitions with the same exponent for h_0 are summed within rows, A001263 is obtained, omitting the leading 1.
From identification of the elements of the inversion with those on page 25 of the Ardila et al. link, the coefficients of the irregular table enumerate non-crossing partitions on [n]. - Tom Copeland, Oct 13 2014
From Tom Copeland, Oct 28-29 2014: (Start)
Operating with d/d(1') = d/d(h_1) on the n-th partition polynomial Prt(n;h_0,h_1,..,h_n) in square brackets above associated with t^(n+1) generates n * Prt(n-1;h_0,h_1,..,h_(n-1)); therefore, the polynomials are an Appell sequence of polynomials in the indeterminate h_1 when h_0=1 (a special type of Sheffer sequence).
Consequently, umbrally, [Prt(.;1,x,h_2,..) + y]^n = Prt(n;1,x+y,h_2,..); that is, Sum_{k=0..n} binomial(n,k) * Prt(k;1,x,h_2,..) * y^(n-k) = Prt(n;1,x+y,h_2,..).
Or, e^(x*z) * exp[Prt(.;1,0,h_2,..) * z] = exp[Prt(.;1,x,h_2,..) * z]. Then with x = h_1 = -(1/2) * d^2[f(t)]/dt^2 evaluated at t=0, the formal Laplace transform from z to 1/t of this expression generates g(t), the comp. inverse of f(t), when h_0 = 1 = df(t)/dt eval. at t=0.
I.e., t / (1 - t*(x + Prt(.;1,0,h_2,..))) = t / (1 - t*Prt(.;1,x,h_2,..)) = g(t), interpreted umbrally, when h_0 = 1.
(End)
Connections to and between arrays associated to the Catalan (A000108 and A007317), Riordan (A005043), Fibonacci (A000045), and Fine (A000957) numbers and to lattice paths, e.g., the Motzkin, Dyck, and Łukasiewicz, can be made explicit by considering the inverse in x of the o.g.f. of A104597(x,-t), i.e., f(x) = P(Cinv(x),t-1) = Cinv(x) / (1 + (t-1)*Cinv(x)) = x*(1-x) / (1 + (t-1)*x*(1-x)) = (x-x^2) / (1 + (t-1)*(x-x^2)), where  Cinv(x) = x*(1-x) is the inverse of C(x) = (1 - sqrt(1-4*x)) / 2, a shifted o.g.f. for the Catalan numbers, and P(x,t) = x / (1+t*x) with inverse Pinv(x,t) = -P(-x,t) = x / (1-t*x). Then h(x,t) = x / f(x,t) = x * (1+(t-1)Cinv(x)) / Cinv(x) = 1 + t*x + x^2 + x^3 + ..., i.e., h_1=t and all other coefficients are 1, so the inverse of f(x,t) in x, which is explicitly in closed form finv(x,t) = C(Pinv(x,t-1)), is given by A091867, whose coefficients are sums of the refined Narayana numbers above obtained by setting h_1=(1')=t in the partition polynomials and all other coefficients to one. The group generators C(x) and P(x,t) and their inverses allow associations to be easily made between these classic number arrays. - Tom Copeland, Nov 03 2014
From Tom Copeland, Nov 10 2014: (Start)
Inverting in x with t a parameter, let F(x;t,n) = x - t*x^(n+1). Then h(x) = x / F(x;t,n) = 1 / (1-t*x^n) = 1 + t*x^n + t^2*x^(2n) + t^3*x^(3n) + ..., so h_k vanishes unless k = m*n with m an integer in which case h_k = t^m.
Finv(x;t,n) = Sum_{j>=0} {binomial((n+1)*j,j) / (n*j + 1)} * t^j * x^(n*j + 1), which gives the Catalan numbers for n=1, and the Fuss-Catalan sequences for n>1 (see A001764, n=2). [Added braces to disambiguate the formula. - N. J. A. Sloane, Oct 20 2015]
This relation reveals properties of the partitions and sums of the coefficients of the array. For n=1, h_k = t^k for all k, implying that the row sums are the Catalan numbers. For n = 2, h_k for k odd vanishes, implying that there are no blocks with only even-indexed h_k on the even-numbered rows and that only the blocks containing only even-sized bins contribute to the odd-row sums giving the Fuss-Catalan numbers for n=2. And so on, for n > 2.
These relations are reflected in any combinatorial structures enumerated by this array and the partitions, such as the noncrossing partitions depicted for a five-element set (a pentagon) in Wikipedia.
(End)
From Tom Copeland, Nov 12 2014: (Start)
An Appell sequence possesses an umbral inverse sequence (cf. A249548). The partition polynomials here, Prt(n;1,h_1,...), are an Appell sequence in the indeterminate h_1=u, so have an e.g.f. exp[Prt(.;1,u,h_2...)*t] = e^(u*t) * exp[Prt(.;1,0,h2,...)*t] with umbral inverses with an e.g.f  e^(-u*t) / exp[Prt(.;1,0,h2,...)*t]. This makes contact with the formalism of A133314 (cf. also A049019 and A019538) and the signed, refined face partition polynomials of the permutahedra (or their duals), which determine the reciprocal of exp[Prt(.,0,u,h2...)*t] (cf. A249548) or exp[Prt(.;1,u,h2,...)*t], forming connections among the combinatorics of permutahedra and the noncrossing partitions, Dyck paths and trees (cf. A125181), and many other important structures isomorphic to the partitions of this entry, as well as to formal cumulants through A127671 and algebraic structures of Lie algebras. (Cf. relationship of permutahedra with the Eulerians A008292.)
(End)
From Tom Copeland, Nov 24 2014: (Start)
The n-th row multiplied by n gives the number of terms in the homogeneous symmetric monomials generated by [x(1) + x(2) + ... + x(n+1)]^n under the umbral mapping x(m)^j = h_j, for any m. E.g., [a + b + c]^2 = [a^2 + b^2 + c^2] + 2 * [a*b + a*c + b*c] is mapped to [3 * h_2] + 2 * [3 * h_1^2], and 3 * A134264(3) = 3 *(1,1)= (3,3) the number of summands in the two homogeneous polynomials in the square brackets. For n=3, [a + b + c + d]^3 = [a^3 + b^3 + ...] + 3 [a*b^2 + a*c^2 + ...] + 6 [a*b*c + a*c*d + ...] maps to  [4 * h_3] + 3 [12 * h_1 * h_2] + 6 [4 * (h_1)^3], and the number of terms in the brackets is given by 4 * A134264(4) = 4 * (1,3,1) = (4,12,4).
The further reduced expression is 4 h_3 + 36 h_1 h_2 + 24 (h_1)^3 = A248120(4) with h_0 = 1. The general relation is n * A134264(n) = A248120(n) / A036038(n-1) where the arithmetic is performed on the coefficients of matching partitions in each row n.
Abramowitz and Stegun give combinatorial interpretations of A036038 and relations to other number arrays.
This can also be related to repeated umbral composition of Appell sequences and topology with the Bernoulli numbers playing a special role. See the Todd class link.
(End)
These partition polynomials are dubbed the Voiculescu polynomials on page 11 of the He and Jejjala link. - Tom Copeland, Jan 16 2015
See page 5 of the Josuat-Verges et al. reference for a refinement of these partition polynomials into a noncommutative version composed of nondecreasing parking functions. - Tom Copeland, Oct 05 2016
(Per Copeland's Oct 13 2014 comment.) The number of non-crossing set partitions whose block sizes are the parts of the n-th integer partition, where the ordering of integer partitions is first by total, then by length, then lexicographically by the reversed sequence of parts. - Gus Wiseman, Feb 15 2019
With h_0 = 1 and the other h_n replaced by suitably signed partition polynomials of A263633, the refined face partition polynomials for the associahedra of normalized A133437 with a shift in indices are obtained (cf. In the Realm of Shadows). - Tom Copeland, Sep 09 2019
Number of primitive parking functions associated to each partition of n. See Lemma 3.8 on p. 28 of Rattan. - Tom Copeland, Sep 10 2019
With h_n = n + 1, the d_k (A006013) of Table 2, p. 18, of Jong et al. are obtained, counting the n-point correlation functions in a quantum field theory. - Tom Copeland, Dec 25 2019
By inspection of the diagrams on Robert Dickau's website, one can see the relationship between the monomials of this entry and the connectivity of the line segments of the noncrossing partitions. - Tom Copeland, Dec 25 2019
Speicher has examples of the first four inversion partition polynomials on pp. 22 and 23 with his k_n equivalent to h_n = (n') here with h_0 = 1. Identifying z = t, C(z) = t/f(t) = h(t), and M(z) = f^(-1)(t)/t, then statement (3), on p. 43, of Theorem 3.26, C(z M(z)) = M(z), is equivalent to substituting f^(-1)(t) for t in t/f(t), and statement (4), M(z/C(z)) = C(z), to substituting f(t) for t in f^(-1)(t)/t. - Tom Copeland, Dec 08 2021
Given a Laurent series of the form f(z) = 1/z + h_1 + h_2 z + h_3 z^2 + ..., the compositional inverse is f^(-1)(z) = 1/z + Prt(1;1,h_1)/z^2 + Prt(2;1,h_1,h_2)/z^3 + ... = 1/z + h_1/z^2 + (h_1^2 + h_2)/z^3 + (h_1^3 + 3 h_1 h_2 + h_3)/z^4 + (h_1^4 + 6 h_1^2 h_2 + 4 h_1 h_3 + 2 h_2^2 + h_4)/z^5 + ... for which the polynomials in the numerators are the partition polynomials of this entry. For example, this formula applied to the q-expansion of Klein's j-invariant / function with coefficients A000521, related to monstrous moonshine, gives the compositional inverse with the coefficients A091406 (see He and Jejjala). - Tom Copeland, Dec 18 2021
The partition polynomials of A350499 'invert' the polynomials of this entry giving the indeterminates h_n. A multinomial formula for the coefficients of the partition polynomials of this entry, equivalent to the multinomial formula presented in the first four sentences of the formula section below, is presented in the MathOverflow question referenced in A350499. - Tom Copeland, Feb 19 2022

			1) With f(t) = t / (t-1), then h(t) = -(1-t), giving h_0 = -1, h_1 = 1 and h_n = 0 for n>1. Then g(t) = -t - t^2 - t^3 - ... = t / (t-1).
2) With f(t) = t*(1-t), then h(t) = 1 / (1-t), giving h_n = 1 for all n. The compositional inverse of this f(t) is g(t) = t*A(t) where A(t) is the o.g.f. for the Catalan numbers; therefore the sum over k of T(j,k), i.e., the row sum, is the Catalan number A000108(j-1).
3) With f(t) = (e^(-a*t)-1) / (-a), h(t) = Sum_{n>=0} Bernoulli(n) * (-a*t)^n / n! and g(t) = log(1-a*t) / (-a) = Sum_{n>=1} a^(n-1) * t^n / n. Therefore with h_n = Bernoulli(n) * (-a)^n / n!, Sum_{permutations s with s(1)+s(2)+...+s(j)=j-1} h_s(1) * h_s(2) * ... * h_s(j) = j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) = a^(j-1). Note, in turn, Sum_{a=1..m} a^(j-1) = (Bernoulli(j,m+1) - Bernoulli(j)) / j for the Bernoulli polynomials and numbers, for j>1.
4) With f(t,x) = t / (x-1+1/(1-t)), then h(t,x) = x-1+1/(1-t), giving (h_0)=x and (h_n)=1 for n>1. Then g(t,x) = (1-(1-x)*t-sqrt(1-2*(1+x)*t+((x-1)*t)^2)) / 2, a shifted o.g.f. in t for the Narayana polynomials in x of A001263.
5) With h(t)= o.g.f. of A075834, but with A075834(1)=2 rather than 1, which is the o.g.f. for the number of connected positroids on [n] (cf. Ardila et al., p. 25), g(t) is the o.g.f. for A000522, which is the o.g.f. for the number of positroids on [n]. (Added Oct 13 2014 by author.)
6) With f(t,x) = x / ((1-t*x)*(1-(1+t)*x)), an o.g.f. for A074909, the reverse face polynomials of the simplices, h(t,x) = (1-t*x) * (1-(1+t)*x) with h_0=1, h_1=-(1+2*t), and h_2=t*(1+t), giving as the inverse in x about 0 the o.g.f. (1+(1+2*t)*x-sqrt(1+(1+2*t)*2*x+x^2)) / (2*t*(1+t)*x) for signed A033282, the reverse face polynomials of the Stasheff polytopes, or associahedra. Cf. A248727. (Added Jan 21 2015 by author.)
7) With f(x,t) = x / ((1+x)*(1+t*x)), an o.g.f. for the polynomials (-1)^n * (1 + t + ... + t^n), h(t,x) = (1+x) * (1+t*x) with h_0=1, h_1=(1+t), and h_2=t, giving as the inverse in x about 0 the o.g.f. (1-(1+t)*x-sqrt(1-2*(1+t)*x+((t-1)*x)^2)) / (2*x*t) for the Narayana polynomials A001263. Cf. A046802. (Added Jan 24 2015 by author.)
From _Gus Wiseman_, Feb 15 2019: (Start)
Triangle begins:
   1
   1
   1   1
   1   3   1
   1   4   2   6   1
   1   5   5  10  10  10   1
   1   6   6   3  15  30   5  20  30  15   1
   1   7   7   7  21  42  21  21  35 105  35  35  70  21   1
Row 5 counts the following non-crossing set partitions:
  {{1234}}  {{1}{234}}  {{12}{34}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
            {{123}{4}}  {{14}{23}}  {{1}{23}{4}}
            {{124}{3}}              {{12}{3}{4}}
            {{134}{2}}              {{1}{24}{3}}
                                    {{13}{2}{4}}
                                    {{14}{2}{3}}
(End)

A. Nica and R. Speicher (editors), Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series: 335, Cambridge University Press, 2006 (see in particular, Eqn. 9.14 on p. 141, enumerating noncrossing partitions).

Andrew Howroyd, Table of n, a(n) for n = 1..2087 (rows 1..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Alexandrov, Enumerative geometry, tau-functions, and Heisenberg-Virasoro algebra, arXiv:hep-th/1404.3402v3, 2015 (p.22).
M. Ashelevich and W. Mlotkowski, Semigroups of distributions with linear Jacobi parameters, arxiv.org/abs/1001.1540v3 [math.CO], p. 9, 2011.
F. Ardila, F. Rincon, and L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698v2 [math.CO], 2013 (p. 25).
T. Banica, S. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, arXiv preprint arXiv:0710.5931 [math.PR], 2008.
Freddy Cachazo and Bruno Giménez Umbert, Connecting Scalar Amplitudes using The Positive Tropical Grassmannian, arXiv:2205.02722 [hep-th], 2022.
David Callan, Sets, Lists and Noncrossing Partitions , arXiv:0711.4841 [math.CO], 2008.
B. Collins, I. Nechita, and K. Zyczkowski, Random graph states, maximal flow and Fuss-Catalan distributions, arXiv preprint arXiv:1003.3075 [quant-ph], 2010.
Tom Copeland, Compositional inversion and Appell sequences, Nov 2, 2014.
Tom Copeland, The Hirzebruch criterion for the Todd class Dec. 14, 2014.
Tom Copeland, Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion Dec. 23, 2014.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Tom Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
Tom Copeland, A Taste of Moonshine in Free Moments, 2022.
R. Dickau, Non-crossing partitions, Robert Dickau's website, 2012.
K. Ebrahimi-Fard, L. Foissy, J. Kock, and F. Patras, Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations, arXiv:1907.01190 [math.CO], 2019, p. 25.
K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, arXiv:1409.5664v2 [math.CO], 2015, p. 12.
K. Ebrahimi-Fard and F. Patras, The splitting process in free probability theory, arXiv:1502.02748 [math.CO], 2015, p. 3.
Y. He and V. Jejjala, Modular Matrix Models, arXiv:hep-th/0307293, 2003.
J. Jong, A. Hock, and R. Wulkenhaar Catalan tables and a recursion relation in noncommutative quantum field theory, arXiv preprint arXiv:1904.11231 [math-ph], 2019.
M. Josuat-Verges, F. Menous, J. Novelli, and J. Thibon, Noncommutative free cumulants, arxiv.org/abs/1604.04759 [math.CO], 2016.
Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
C. Lenart, Lagrange inversion and Schur functions, Journal of Algebraic Combinatorics, Vol. 11, Issue 1, p. 69-78, 2000, (see p. 70, Eqn. 1.2).
M. Mastnak and A. Nica, Hopf algebras and the logarithm of the S-transform in free probability, arXiv:0807.4169v2 [math.OA], p. 28, 2008-2009.
MathOverflow, Guises of the Noncrossing Partitions (NCPs), an MO question posed by Tom Copeland, 2019.
MathOverflow, Combinatorial/diagrammatic models for free moment partition polynomials, an MO question posed by Tom Copeland, 2021.
J. McCammond, Non-crossing Partitions in Surprising Locations, American Mathematical Monthly 113 (2006) 598-610.
M. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], 2016, pp. 33-34 Example 10.
J. Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, arXiv:math/9908029 [math.CO], 1999.
A. Rattan, Parking functions and related combinatorial structures, Master's thesis, Univ. of Waterloo, 2014.
A. Schuetz and G. Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
R. Simion, Noncrossing partitions, Discrete Mathematics, Vol. 217, Issues 1-3, pp. 367-409, 2000.
R. Speicher, Lecture Notes on "Free Probability Theory", arXiv:1908.08125 [math.OA], 2019.
R. Stanley, Parking Functions and Noncrossing Partitions, 1996.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
Wikipedia, Noncrossing partition
J. Zhou, On emergent geometry of the Gromov-Witten theory of quintic Calabi-Yau threefold, arXiv:2008.03407 [math-ph], 2020.
Index entries for sequences related to Łukasiewicz

(A001263,A119900) = (reduced array, associated g(x)). See A145271 for meaning and other examples of reduced and associated.
Other orderings are A125181 and A306438.
Cf. A119900 (e.g.f. for reduced W(x) with (h_0)=t and (h_n)=1 for n>0).
Cf. A248927 and A248120, "scaled" versions of this Lagrange inversion.
Cf. A091867 and A125181, for relations to lattice paths and trees.
Cf. A000045, A000108, A000957, A001764, A000522, A005043, A007317, A033282,A036038, A046802, A074909, A075834, A104597, A145271, A248727.
Cf. A249548 for use of Appell properties to generate the polynomials.
Cf. A133314, A049019, A019538, A127671, and A008292 for relations to permutahedra, Eulerians.
Cf. A263634, A263633, A000041, A000110, A001263, A016098, A124794, A133437.
Cf. A006013.
Cf. A000521, A091406.
Cf. A350499

Table[Binomial[Total[y],Length[y]-1]*(Length[y]-1)!/Product[Count[y,i]!,{i,Max@@y}],{n,7},{y,Sort[Sort/@IntegerPartitions[n]]}] (* Gus Wiseman, Feb 15 2019 *)

C(v)={my(n=vecsum(v), S=Set(v)); n!/((n-#v+1)!*prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!))}
row(n)=[C(Vec(p)) | p<-partitions(n-1)]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

For j>1, there are P(j,m;a...) = j! / [ (j-m)! (a_1)! (a_2)! ... (a_(j-1))! ] permutations of h_0 through h_(j-1) in which h_0 is repeated (j-m) times; h_1, repeated a_1 times; and so on with a_1 + a_2 + ... + a_(j-1) = m.
If, in addition, a_1 + 2 * a_2 + ... + (j-1) * a_(j-1) = j-1, then each distinct combination of these arrangements is correlated with a partition of j-1.
T(j,k) is [ P(j,m;a...) / j ] for the k-th partition of j-1 as described in the comments.
For example from g(t) above, T(5,4) = (5! / ((5-3)! * 2!)) / 5 = 6 for the 4th partition under n=5-1=4 with m=3 parts in A&S.
From Tom Copeland, Sep 30 2011: (Start)
Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}
  = [(h_0)-1+:1/(1-h.*x):]^2 / {(h_0)-:[h.x/(1-h.x)]^2:}
  = [(h_0)+(h_1)x+(h_2)x^2+...]^2 / [(h_0)-(h_2)x^2-2(h_3)x^3-3(h_4)x^4-...], where :" ": denotes umbral evaluation of the expression within the colons and h. is an umbral coefficient.
Then for the partition polynomials of A134264,
  Poly[n;h_0,...,h_(n-1)]=(1/n!)(W(x)*d/dx)^n x, evaluated at x=0, and the compositional inverse of f(t) is g(t) = exp(t*W(x)*d/dx) x, evaluated at x=0. Also, dg(t)/dt = W(g(t)), and g(t) gives A001263 with (h_0)=u and (h_n)=1 for n>0 and A000108 with u=1.
(End)
From Tom Copeland, Oct 20 2011: (Start)
With exp(x* PS(.,t)) = exp(t*g(x)) = exp(x*W(y)d/dy) exp(t*y) eval. at y=0, the raising (creation) and lowering (annihilation) operators defined by R PS(n,t) = PS(n+1,t) and L PS(n,t) = n*PS(n-1,t) are
R = t*W(d/dt) = t*((h_0) + (h_1)d/dt + (h_2)(d/dt)^2 + ...)^2 / ((h_0) - (h_2)(d/dt)^2 - 2(h_3)(d/dt)^3 - 3(h_4)(d/dt)^4 + ...),  and
L = (d/dt)/h(d/dt) = (d/dt) 1/((h_0) + (h_1)*d/dt + (h_2)*(d/dt)^2 + ...)
Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0 are the row polynomials of A134264. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.)
(End)
Using the formalism of A263634, the raising operator for the partition polynomials of this array with h_0 = 1 begins as R = h_1 + h_2 D + h_3 D^2/2! + (h_4 - h_2^2) D^3/3! + (h_5 - 5 h_2 h_3) D^4/4! + (h_6 + 5 h_2^3 - 7 h_3^2 - 9 h_2 h_4) D^5/5! + (h_7 - 14 h_2 h_5 + 56 h_2^2 h_3) D^6/6! + ... with D = d/d(h_1). - Tom Copeland, Sep 09 2016
Let h(x) = x/f^{-1}(x) = 1/[1-(c_2*x+c_3*x^2+...)], with c_n all greater than zero. Then h_n are all greater than zero and h_0 = 1. Determine P_n(t) from exp[t*f^{-1}(x)] = exp[x*P.(t)] with f^{-1}(x) = x/h(x) expressed in terms of the h_n (cf. A133314 and A263633). Then P_n(b.) = 0 gives a recursion relation for the inversion polynomials of this entry a_n = b_n/n! in terms of the lower order inversion polynomials and P_j(b.)P_k(b.) = P_j(t)P_k(t)|{t^n = b_n} = d{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)]. - Tom Copeland, Feb 09 2018
A raising operator for the partition polynomials with h_0 = 1 regarded as a Sheffer Appell sequence in h_1 is described in A249548. - Tom Copeland, Jul 03 2018

Added explicit t^6, t^7, and t^8 polynomials and extended initial table to include the coefficients of t^8. - Tom Copeland, Sep 14 2016
Title modified by Tom Copeland, May 28 2018
More terms from Gus Wiseman, Feb 15 2019
Title modified by Tom Copeland, Sep 10 2019

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!).

A318762 Number of permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 6, 6, 4, 1, 12, 1, 5, 10, 24, 1, 30, 1, 20, 15, 6, 1, 60, 20, 7, 90, 30, 1, 60, 1, 120, 21, 8, 35, 180, 1, 9, 28, 120, 1, 105, 1, 42, 210, 10, 1, 360, 70, 140, 36, 56, 1, 630, 56, 210, 45, 11, 1, 420, 1, 12, 420, 720, 84, 168, 1, 72, 55

Offset: 1

Gus Wiseman, Sep 03 2018

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(12) = 12 permutations are (1123), (1132), (1213), (1231), (1312), (1321), (2113), (2131), (2311), (3112), (3121), (3211).

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000041, A000110, A000258, A000670, A005651, A008277, A008480, A056239, A112624, A112798, A124794, A215366, A300335, A305936.

a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i->
       numtheory[pi](i[1])$i[2], ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 03 2018

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[primeMS[n]]!/Times@@Factorial/@primeMS[n],{n,100}]

sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
a(n)={if(n==1, 1, my(s=sig(n)); vecsum(s)!/prod(i=1, #s, s[i]!))}  \\ Andrew Howroyd, Dec 17 2018

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) = (Sum x_i * y_i)! / Product (x_i!)^y_i.
a(n) = A008480(A181821(n)).
a(n) = A112624(n) * A124794(n). - Max Alekseyev, Oct 15 2023
Sum_{m in row n of A215366} a(m) = A005651(n).
Sum_{m in row n of A215366} a(m) * A008480(m) = A000670(n).
Sum_{m in row n of A215366} a(m) * A008480(m) / A001222(m)! = A000110(n).

A123023 a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0.

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875

Offset: 0

Roger L. Bagula, Sep 24 2006

a(n) is the number of ways of separating n terms into pairs. - Stephen Crowley, Apr 07 2007
a(n) is the n-th moment of the standard normal distribution. - Hal M. Switkay, Nov 06 2019
a(n) is the number of fixed-point free involutions in the symmetric group of degree n. - Nick Krempel, Feb 26 2020

			From _Gus Wiseman_, Dec 23 2018: (Start)
The a(6) = 15 ways of partitioning {1,2,3,4,5,6} into disjoint pairs:
  {{12}{34}{56}},  {{12}{35}{46}},  {{12}{36}{45}},
  {{13}{24}{56}},  {{13}{25}{46}},  {{13}{26}{45}},
  {{14}{23}{56}},  {{14}{25}{36}},  {{14}{26}{35}},
  {{15}{23}{46}},  {{15}{24}{36}},  {{15}{26}{34}},
  {{16}{23}{45}},  {{16}{24}{35}},  {{16}{25}{34}}.
(End)

Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York, 1973, page 107, solved problem 19.18
Norbert Wiener, Nonlinear Problems in Random Theory, 1958, Equation 1.31

G. C. Greubel, Table of n, a(n) for n = 0..799
Mihai Nica, Terence Tao's central limit theorem, Double Factorials (!!) and the Moment Method, YouTube video (2023).
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018, Table 2 on p. 15.

Cf. A000085, A000110, A000124, A000142, A000587, A001147, A001464, A006129, A079267, A124794, A186021, A295193.

a:=[1,0]; [n le 2 select a[n] else (n-2)*Self(n-2): n in [1..30]]; // Marius A. Burtea, Nov 07 2019

with(combstruct): ZL2 := [S, {S=Set(Cycle(Z, card=2))}, labeled]:
seq(count(ZL2, size=n), n=0..36); # Zerinvary Lajos, Sep 24 2007
a := n -> ifelse(irem(n, 2) = 1, 0, 2^(n/2) * pochhammer(1/2, n/2)):
seq(a(n), n = 0..36); # Peter Luschny, Jan 11 2023

RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == (n - 1) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *)

a(n) = (1/2)*Gamma((1/2)*n + 1/2)*2^((1/2)*n)*(1 + (-1)^n)/sqrt(Pi). - Stephen Crowley, Apr 07 2007
E.g.f.: exp(x^2/2). - Geoffrey Critzer, Mar 15 2009
a(2n) = A001147(n). - R. J. Mathar, Oct 11 2011
From Sergei N. Gladkovskii, Nov 18 2012, Dec 05 2012, May 16 2013, May 24 2013, Jun 07 2013: (Start)
Continued fractions:
E.g.f.: E(0) where E(k) = 1 + x^2*(4*k+1)/((4*k+2)*(4*k+3) - x^2*(4*k+2)*(4*k+3)^2/(x^2*(4*k+3) + (4*k+4)*(4*k+5)/E(k+1))).
G.f.: 1/G(0) where G(k) =  1 - x^2*(k+1)/G(k+1).
G.f.: 1 + x^2/(1+x) + Q(0)*x^3/(1+x), where Q(k) = 1 + (2*k+3)*x/(1 - x/(x + 1/Q(k+1))).
G.f.: G(0)/2, where G(k) = 1 + 1/(1-x/(x+1/x/(2*k+1)/G(k+1))).
G.f.: (G(0) - 1)*x/(1+x) + 1, where G(k) = 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))). (End)
For n even, a(n) = A001147(n/2) = A124794(3^(n/2)). a(n) is also the coefficient of x1*...*xn in Product_{1 <= i < j <= n} (1 + xi*xj). - Gus Wiseman, Dec 23 2018
a(n) = 2^(n/2)*Pochhammer(1/2, n/2)*(n+1 mod 2). - Peter Luschny, Jan 11 2023

Edited by N. J. A. Sloane, Jan 06 2008
Better name by Sergei N. Gladkovskii, May 24 2013
Leading term 1 dropped, offset changed, and entry edited correspondingly by Andrey Zabolotskiy, Nov 07 2019

A321912 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in e(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 3, 1, 3, 6, 0, 0, 0, 0, 1, 0, 1, 0, 2, 6, 0, 0, 0, 1, 4, 0, 2, 1, 5, 12, 1, 6, 4, 12, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0, 0, 1, 0, 3, 10, 0, 0, 1, 5, 2, 12, 30, 0, 0, 0, 2, 1, 7, 20, 0, 1, 3, 12, 7, 27, 60, 1, 5

Offset: 1

Gus Wiseman, Nov 22 2018

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

Also the coefficient of f(v) in h(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):      1
  (11):  1  2
.
  (3):          1
  (21):      1  3
  (111):  1  3  6
.
  (4):                 1
  (22):       1     2  6
  (31):             1  4
  (211):      2  1  5 12
  (1111):  1  6  4 12 24
.
  (5):                        1
  (41):                    1  5
  (32):              1     3 10
  (221):          1  5  2 12 30
  (311):             2  1  7 20
  (2111):      1  3 12  7 27 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: e(32) = m(221) + 3m(2111) + 10m(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321742.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321913-A321935.

A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 2, -3, 1, 6, 3, 8, 6, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 2, -1, 0, -2, 1, -6, 3, 8, -6, 1, 24, 30, 20, 15, 20, 10, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, 0, 6, -4, 3, -4, -2, 1, 4, 0, 0, -5, 0, 0, 1, -6, 0, 5, 0, 5

Offset: 1

Gus Wiseman, Nov 23 2018

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

We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is the basis of Schur functions and syt(y) is the number of standard Young tableaux of shape y.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1  1
  (11): -1  1
.
  (3):    2  3  1
  (21):  -1     1
  (111):  2 -3  1
.
  (4):     6  3  8  6  1
  (22):       3 -4     1
  (31):   -2 -1     2  1
  (211):   2 -1    -2  1
  (1111): -6  3  8 -6  1
.
  (5):     24 30 20 15 20 10  1
  (41):    -6    -5     5  5  1
  (32):       -6  4  3 -4  2  1
  (221):       6 -4  3 -4 -2  1
  (311):    4       -5        1
  (2111):  -6     5     5 -5  1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321900.

Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.

A306438 Number of non-crossing set partitions whose block sizes are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 4, 1, 6, 1, 5, 5, 1, 1, 10, 1, 10, 6, 6, 1, 10, 3, 7, 5, 15, 1, 30, 1, 1, 7, 8, 7, 30, 1, 9, 8, 20, 1, 42, 1, 21, 21, 10, 1, 15, 4, 21, 9, 28, 1, 35, 8, 35, 10, 11, 1, 105, 1, 12, 28, 1, 9, 56, 1, 36, 11, 56, 1, 70, 1, 13, 28, 45, 9

Offset: 1

Gus Wiseman, Feb 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(18) = 10 non-crossing set partitions of type (2, 2, 1) are:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
  {{1,2},{3},{4,5}}
  {{1,2},{3,4},{5}}
  {{1,2},{3,5},{4}}
  {{1,3},{2},{4,5}}
  {{1,4},{2,3},{5}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}
Missing from this list are the following crossing set partitions:
  {{1},{2,4},{3,5}}
  {{1,3},{2,4},{5}}
  {{1,3},{2,5},{4}}
  {{1,4},{2},{3,5}}
  {{1,4},{2,5},{3}}

Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
Wikipedia, Noncrossing partition.

Other orderings are A125181 and A134264.

Cf. A000041, A000108, A000110, A000670, A001263, A001764, A008480, A056239, A112798, A124794, A306437.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,1,With[{y=primeMS[n]},Binomial[Total[y],Length[y]-1]*(Length[y]-1)!/Product[Count[y,i]!,{i,Max@@y}]]],{n,80}]

a(n) = falling(m, k - 1)/Product_i (y)_i! where m is the sum of parts (A056239(n)), k is the number of parts (A001222(n)), y is the integer partition with Heinz number n (row n of A296150), (y)_i is the number of i's in y, and falling(x, y) is the falling factorial x(x - 1)(x - 2) ... (x - y + 1) [Kreweras].

Equivalently, a(n) = falling(A056239(n), A001222(n) - 1)/A112624(n).

A319225 Number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 13 2018

nonn

a(1) = 1 by convention.

A prime index of n is a number m such that prime(m) divides n.

			Of the cycle ({1,2,3}, {(1,2),(2,3),(3,1)}) the spanning subgraphs where the sizes of connected components are (2,1) are: ({1,2,3}, {(1,2)}), ({1,2,3}, {(2,3)}), ({1,2,3}, {(3,1)}). Since the prime indices of 6 are (2,1), we conclude a(6) = 3.

Wikipedia, Expressing power sums in terms of elementary symmetric polynomials
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Different orderings with signs are A115131, A210258, A263916.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A215366, A318762, A319191, A319193, A319226.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[With[{m=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Select[Subsets[Partition[Range[Total[m]],2,1,1],{Total[m]-PrimeOmega[n]}],Sort[Length/@csm[Union[#,List/@Range[Total[m]]]]]==m&]]],{n,30}]

a(n) = A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A321742 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A318762 Number of permutations of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A123023 a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A321912 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in e(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

Views

Author

Keywords

Comments

Examples