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A341105
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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).
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0
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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, 720, 7, 6, 10, 10, 12, 8, 24, 18, 24, 12, 72, 48, 48, 240, 5040, 8, 7, 12, 12, 15, 10, 30, 32, 12, 32, 16, 96, 36, 36, 24, 36, 360, 384, 96, 192, 1440, 40320
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OFFSET
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0,3
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COMMENTS
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By the 'standard order' of partitions we understand the graded reverse lexicographic ordering A080577.
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.
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LINKS
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FORMULA
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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].
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EXAMPLE
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Triangle begins:
[0] [1]
[1] [1]
[2] [2, 2]
[3] [3, 2, 6]
[4] [4, 3, 8, 4, 24]
[5] [5, 4, 6, 6, 8, 12, 120]
[6] [6, 5, 8, 8, 18, 6, 18, 48, 16, 48, 720]
[7] [7, 6, 10, 10, 12, 8, 24, 18, 24, 12, 72, 48, 48, 240, 5040]
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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].
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PROG
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(SageMath)
def PartitionsFreq(n): # returns a generator object
return ([sum((1 if v == m else 0) for j, v in enumerate(p)) for m in (1..n)]
for p in Partitions(n))
def A341105row(n): # returns a generator object
return (product(factorial(p[i])*(i+1)^p[i] for i in range(n))
for p in PartitionsFreq(n))
for n in range(9): print(list(A341105row(n)))
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CROSSREFS
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The row terms are a permutation of the row terms of A110141.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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