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A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

%I A383095 #10 Apr 26 2025 11:26:50
%S A383095 1,1,2,2,3,2,6,2,4,5,6,2,12,2,6,8,5,2,20,2,12,8,6,2,20,5,6,12,12,2,34,
%T A383095 2,6,8,6,8,45,2,6,8,20,2,34,2,12,28,6,2,30,5,20,8,12,2,52,8,20,8,6,2,
%U A383095 78,2,6,28,7,8,34,2,12,8,34,2,80,2,6,28,12,8,34,2,30,25
%N A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.
%e A383095 The partition (2,2,1,1) has permutation (2,1,1,2) so is counted under a(6).
%e A383095 The a(1) = 1 through a(10) = 6 partitions (A=10):
%e A383095   1  2   3    4     5      6       7        8         9          A
%e A383095      11  111  22    11111  33      1111111  44        333        55
%e A383095               1111         222              2222      33111      22222
%e A383095                            2211             11111111  3111111    2221111
%e A383095                            21111                      111111111  22111111
%e A383095                            111111                                1111111111
%t A383095 Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Total/@Split[#]&]]==1&]],{n,0,15}]
%Y A383095 For distinct instead of equal run-sums we have A000005.
%Y A383095 For run-lengths instead of sums we have A383094.
%Y A383095 The complement is counted by A383096 + A383097, ranks A383100 \/ A383015.
%Y A383095 These partitions are ranked by A383099 = positions of 1 in A382877.
%Y A383095 Counting and ranking partitions by run-lengths and run-sums:
%Y A383095 - constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
%Y A383095 - distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
%Y A383095 - weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
%Y A383095 - weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
%Y A383095 - strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
%Y A383095 - strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
%Y A383095 A275870 counts collapsible partitions, ranks A300273.
%Y A383095 A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
%Y A383095 A353851 counts compositions with all equal run-sums, ranks A353848.
%Y A383095 A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
%Y A383095 Cf. A329738, A353839, A353832, A353932, A354584, A381636, A381871, A382076, A382876.
%K A383095 nonn
%O A383095 0,3
%A A383095 _Gus Wiseman_, Apr 16 2025
%E A383095 More terms from _Bert Dobbelaere_, Apr 26 2025