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A380401 Triangle read by rows: T(n,k) is the number of necklace permutations of a multiset whose multiplicities are given by the k-th partition of n in graded reflected lexicographic order.

%I A380401 #29 Jan 26 2025 14:18:00
%S A380401 1,1,1,2,1,1,6,3,2,1,1,24,12,6,4,2,1,1,120,60,30,16,20,10,4,5,3,1,1,
%T A380401 720,360,180,90,120,60,30,20,30,15,5,6,3,1,1,5040,2520,1260,630,318,
%U A380401 840,420,210,140,70,210,105,54,35,10,42,21,7,7,4,1,1,40320,20160,10080,5040,2520,6720,3360,1680,840,1120,560,188,1680,840,420,280,140,70,336,168,84,56,14,56,28,10,8,4,1,1
%N A380401 Triangle read by rows: T(n,k) is the number of necklace permutations of a multiset whose multiplicities are given by the k-th partition of n in graded reflected lexicographic order.
%C A380401 See A318810 for a definition of necklace permutation.
%D A380401 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, pages 36-37, 42-43.
%H A380401 Math StackExchange, Marko Riedel et. al, <a href="https://math.stackexchange.com/questions/5025780/">Free circular permutations</a>
%H A380401 Marko Riedel, <a href="/A380401/a380401.maple.txt">Maple code for sequence by PET and closed form.</a>
%F A380401 For a distribution of colors n1+n2+...+nm = n the number of necklaces is (1/n)*Sum_{d|gcd(n1,n2,...,nm)} phi(d) (n/d)!/Prod_{q=1..m} (nq/d)!
%F A380401 T(n,k) = A318810(A334434(n,k)).
%e A380401 The ordering of the partitions used here is graded reflected lexicographic illustrated below with n=5:
%e A380401   1,1,1,1,1 => 24
%e A380401   1,1,1,2 => 12
%e A380401   1,2,2 => 6
%e A380401   1,1,3 => 4
%e A380401   2,3 => 2
%e A380401   1,4 => 1
%e A380401   5 => 1
%e A380401 Table begins:
%e A380401   1
%e A380401   1,1
%e A380401   2,1,1
%e A380401   6,3,2,1,1
%e A380401   24,12,6,4,2,1,1
%o A380401 (PARI)
%o A380401 C(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, eulerphi(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
%o A380401 Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)]))} \\ _Andrew Howroyd_, Jan 23 2025
%Y A380401 Cf. A000041 (row lengths), A072605 (row sums), A080576 (graded reflected lexicographic order), A212359 (similar triangle for Abramowitz-Stegun order), A318810, A334434, A214609 (up to rotations and reflections).
%K A380401 nonn,tabf
%O A380401 1,4
%A A380401 _Marko Riedel_, Jan 23 2025