Revision History for A364536
(Underlined text is an addition;
strikethrough text is a deletion.)
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#9 by Andrew Howroyd at Tue Sep 26 13:39:09 EDT 2023
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#8 by Michel Marcus at Tue Sep 26 12:54:34 EDT 2023
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#7 by Chai Wah Wu at Tue Sep 26 12:53:43 EDT 2023
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#6 by Chai Wah Wu at Tue Sep 26 12:53:38 EDT 2023
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| PROG
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(Python)
from collections import Counter
from sympy.utilities.iterables import partitions
def A364536(n): return sum(1 for s, p in map(lambda x: (x[0], tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(), default=1)==1, partitions(n, size=True))) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
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approved
editing
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#5 by Michael De Vlieger at Mon Jul 31 21:47:53 EDT 2023
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#4 by Gus Wiseman at Mon Jul 31 19:56:18 EDT 2023
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#3 by Gus Wiseman at Mon Jul 31 19:56:02 EDT 2023
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| NAME
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Number of strict integer partitions of n where some part withis partsa notdifference disjointof fromtwo firstconsecutive differencesparts.
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| COMMENTS
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In other words, somestrict partpartitions iswith aparts differencenot ofdisjoint twofrom consecutivefirst partsdifferences.
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| CROSSREFS
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For all differences of pairs parts we have A363226 (complement A364346), , non-strict A363225 (complement A364345)..
For all non-differences of pairs we have A364346, strict A364345.
For subsets of {1..n} instead of partitions we have A364466, complement A364463.
The non-strict case is A364467, complementranks A363260A364537.
A323092 counts double-free partitions, ranks A320340, subsets A050291 (complementstrict A088808).A120641.
A325325 counts partitions with distinct first-differences, strict A320347.
Cf. A002865, A025065, `, A093971, A108917, `A196723, A229816, A236912, A237113, A237667, A320347, A326083, A363225, A364347.
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#2 by Gus Wiseman at Mon Jul 31 04:13:14 EDT 2023
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| NAME
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allocatedNumber of strict integer partitions of n with parts not disjoint forfrom Gusfirst Wisemandifferences.
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| DATA
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0, 0, 0, 1, 0, 0, 2, 1, 2, 2, 5, 4, 6, 6, 9, 11, 16, 17, 23, 25, 30, 38, 48, 55, 65, 78, 92, 106, 127, 146, 176, 205, 230, 277, 315, 366, 421, 483, 552, 640, 727, 829, 950, 1083, 1218, 1408, 1577, 1794, 2017, 2298, 2561, 2919, 3255, 3685, 4116, 4638, 5163
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| OFFSET
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0,7
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| COMMENTS
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In other words, some part is a difference of two consecutive parts.
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| EXAMPLE
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The a(3) = 1 through a(15) = 11 partitions (A = 10, B = 11, C = 12):
21 . . 42 421 431 63 532 542 84 742 743 A5
321 521 621 541 632 642 841 752 843
631 821 651 A21 761 942
721 5321 921 5431 842 C21
4321 5421 6421 B21 6432
6321 7321 6431 6531
6521 7431
7421 7521
8321 8421
9321
54321
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| MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, -Differences[#]]!={}&]], {n, 0, 30}]
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| CROSSREFS
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For all differences of pairs parts we have A363226 (complement A364346), non-strict A363225 (complement A364345).
The strict complement is counted by A364464, non-strict A363260.
For subsets of {1..n} instead of partitions we have A364466, complement A364463.
The non-strict case is A364467, complement A363260.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340, subsets A050291 (complement A088808).
A325325 counts partitions with distinct first-differences.
Cf. A002865, A025065, `A093971, A108917, `A196723, A229816, A236912, A237113, A237667, A320347, A326083, A363225, A364347.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Jul 31 2023
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Thu Jul 27 21:46:07 EDT 2023
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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