Revision History for A333144
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| A333144
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Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.
(history;
published version)
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#5 by Peter Luschny at Fri Apr 10 14:39:55 EDT 2020
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#4 by Peter Luschny at Fri Apr 10 11:14:11 EDT 2020
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#3 by Peter Luschny at Fri Apr 10 10:20:34 EDT 2020
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| EXAMPLE
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[0] [1] [1] [1] [2] [1, 2] [3] [1, 1, 6] [4] [1, 1, 2, 2, 24] [5] [1, 1, 1, 2, 2, 6, 120] [6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720] [7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]
[0] [1]
[1] [1]
[2] [1, 2]
[3] [1, 1, 6]
[4] [1, 1, 2, 2, 24]
[5] [1, 1, 1, 2, 2, 6, 120]
[6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720]
[7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]
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Discussion
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| Fri Apr 10
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| Peter Luschny: These factors appear for example in the computation of the Kreweras numbers.
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#2 by Peter Luschny at Fri Apr 10 10:19:28 EDT 2020
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| NAME
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allocatedIrregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated forin Petercanonical Luschnyorder.
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| DATA
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1, 1, 1, 2, 1, 1, 6, 1, 1, 2, 2, 24, 1, 1, 1, 2, 2, 6, 120, 1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040, 1, 1, 1, 2, 1, 1, 6, 2, 1, 2, 2, 24, 2, 4, 2, 6, 120, 24, 12, 48, 720, 40320
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| OFFSET
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0,4
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| COMMENTS
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By 'canonical order' we understand the graded reverse lexicographic order (the default order of Mathematica and SageMath).
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| EXAMPLE
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The irregular table starts:
[0] [1] [1] [1] [2] [1, 2] [3] [1, 1, 6] [4] [1, 1, 2, 2, 24] [5] [1, 1, 1, 2, 2, 6, 120] [6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720] [7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]
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| PROG
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(SageMath)
def A333144row(n):
return [product(factorial(expo) for expo in partition.to_exp()) for partition in Partitions(n)]
for n in (0..9): print(A333144row(n))
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| CROSSREFS
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Row sums are A161779.
Cf. A069123.
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| KEYWORD
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allocated
nonn,tabf
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| AUTHOR
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Peter Luschny, Apr 10 2020
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| STATUS
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approved
editing
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#1 by Peter Luschny at Mon Mar 09 06:21:24 EDT 2020
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| NAME
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allocated for Peter Luschny
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| KEYWORD
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allocated
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| STATUS
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approved
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