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 A306438 Number of non-crossing set partitions whose block sizes are the prime indices of n. 16
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS 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. LINKS Germain Kreweras, Sur les partitions non croisÃ©es d'un cycle, Discrete Math. 1 333-350 (1972). Wikipedia, Noncrossing partition. FORMULA 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). EXAMPLE 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}} MATHEMATICA 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}] CROSSREFS Other orderings are A125181 and A134264. Cf. A000041, A000108, A000110, A000670, A001263, A001764, A008480, A056239, A112798, A124794, A306437. Sequence in context: A065836 A078712 A287218 * A221978 A035254 A143934 Adjacent sequences:  A306435 A306436 A306437 * A306439 A306440 A306441 KEYWORD nonn AUTHOR Gus Wiseman, Feb 15 2019 STATUS approved

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Last modified January 27 13:45 EST 2022. Contains 350607 sequences. (Running on oeis4.)