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A341105 T(n, k) is the Cauchy coefficient of the k-th partition of n, where the partitions are enumerated in standard order. T(n, k) for n >= 0 and 1 <= k <= A000041(n).

%I A341105 #12 Jul 15 2021 21:27:07
%S A341105 1,1,2,2,3,2,6,4,3,8,4,24,5,4,6,6,8,12,120,6,5,8,8,18,6,18,48,16,48,
%T A341105 720,7,6,10,10,12,8,24,18,24,12,72,48,48,240,5040,8,7,12,12,15,10,30,
%U A341105 32,12,32,16,96,36,36,24,36,360,384,96,192,1440,40320
%N A341105 T(n, k) is the Cauchy coefficient of the k-th partition of n, where the partitions are enumerated in standard order. T(n, k) for n >= 0 and 1 <= k <= A000041(n).
%C A341105 By the 'standard order' of partitions we understand the graded reverse lexicographic ordering A080577.
%C A341105 We call the coefficients the 'Cauchy coefficients' because they were used by Cauchy in his proof of the number of permutations on [n] with cycle structure p.
%F A341105 Let p be the k-th partition of n with frequency vector f. Then T(n, k) = Product_{i=1..n} f[i]! * i^f[i].
%e A341105 Triangle begins:
%e A341105 [0] [1]
%e A341105 [1] [1]
%e A341105 [2] [2, 2]
%e A341105 [3] [3, 2, 6]
%e A341105 [4] [4, 3, 8, 4, 24]
%e A341105 [5] [5, 4, 6, 6, 8, 12, 120]
%e A341105 [6] [6, 5, 8, 8, 18, 6, 18, 48, 16, 48, 720]
%e A341105 [7] [7, 6, 10, 10, 12, 8, 24, 18, 24, 12, 72, 48, 48, 240, 5040]
%e A341105 .
%e A341105 For instance, the 40th partition of n = 12 is [5, 2, 2, 2, 1], and has the frequency vector [1, 3, 0, 0, 1]. Thus T(12, 40) = (1!*1^1)*(3!*2^3)*(1!*5^1) = 240. To compute this value with the Sage program below invoke list(A341105row(12))[40].
%o A341105 (SageMath)
%o A341105 def PartitionsFreq(n): # returns a generator object
%o A341105     return ([sum((1 if v == m else 0) for j, v in enumerate(p)) for m in (1..n)]
%o A341105             for p in Partitions(n))
%o A341105 def A341105row(n): # returns a generator object
%o A341105     return (product(factorial(p[i])*(i+1)^p[i] for i in range(n))
%o A341105             for p in PartitionsFreq(n))
%o A341105 for n in range(9): print(list(A341105row(n)))
%Y A341105 The row terms are a permutation of the row terms of A110141.
%Y A341105 Cf. A000041, A080577.
%K A341105 nonn,tabf
%O A341105 0,3
%A A341105 _Peter Luschny_, Feb 25 2021