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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).
0
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
OFFSET
0,3
COMMENTS
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.
FORMULA
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].
EXAMPLE
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]
.
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].
PROG
(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)))
CROSSREFS
The row terms are a permutation of the row terms of A110141.
Sequence in context: A078224 A159688 A128710 * A290309 A095757 A144368
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Feb 25 2021
STATUS
approved