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

0,3

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.

Links

Table of n, a(n) for n=0..66.

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.

Cf. A000041, A080577.

Sequence in context: A078224 A159688 A128710 * A290309 A095757 A144368

Adjacent sequences: A341102 A341103 A341104 * A341106 A341107 A341108

Keyword

nonn,tabf

Author

Peter Luschny, Feb 25 2021

Status

approved