This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383097 #9 Apr 26 2025 11:26:57
%S A383097 0,0,0,0,1,0,1,0,3,0,1,0,7,0,1,0,9,0,7,0,12,0,1,0,38,0,1,1,18,0,38,0,
%T A383097 32,0,1,0,90,0,1,0,71,0,78,0,33,10,1,0,228,0,31,0,42,0,156,0,123,0,1,
%U A383097 0,447,0,1,16,146,0,222,0,63,0,102,0,811,0,1,29,75,0,334,0
%N A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.
%e A383097 The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
%e A383097 The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
%e A383097 (211) (3111) (422) (511111) (633) (71111111) (844)
%e A383097 (41111) (6222) (82222)
%e A383097 (221111) (33222) (442222)
%e A383097 (4221111) (44221111)
%e A383097 (6111111) (422221111)
%e A383097 (33111111) (811111111)
%e A383097 (222111111) (4411111111)
%e A383097 (42211111111)
%e A383097 (222211111111)
%t A383097 Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]
%Y A383097 These partitions are ranked by A383015, positions of terms > 1 in A382877.
%Y A383097 For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
%Y A383097 The complement is A383095 + A383096, ranks A383099 \/ A383100.
%Y A383097 For any positive number of permutations we have A383098, ranks A383110.
%Y A383097 Counting and ranking partitions by run-lengths and run-sums:
%Y A383097 - constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
%Y A383097 - distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
%Y A383097 - weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
%Y A383097 - weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
%Y A383097 - strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
%Y A383097 - strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
%Y A383097 A275870 counts collapsible partitions, ranks A300273.
%Y A383097 A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
%Y A383097 A353851 counts compositions with all equal run-sums, ranks A353848.
%Y A383097 A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
%Y A383097 Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.
%K A383097 nonn
%O A383097 0,9
%A A383097 _Gus Wiseman_, Apr 17 2025
%E A383097 More terms from _Bert Dobbelaere_, Apr 26 2025