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).

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

Author

Peter Luschny, Feb 25 2021

Keywords

• nonn
• tabf

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.

Examples

			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].
		

Crossrefs

The row terms are a permutation of the row terms of A110141.

Cf. A000041, A080577.

Programs

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)))

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].