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A352833 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.

%I A352833 #15 Jul 09 2025 04:58:42
%S A352833 1,0,0,1,1,1,2,1,3,2,4,3,6,5,8,7,12,10,16,14,23,19,30,26,42,35,54,47,
%T A352833 73,62,94,82,124,107,158,139,206,179,260,230,334,293,420,372,532,470,
%U A352833 664,591,835,740,1034,924,1288,1148,1588,1422,1962,1756,2404,2161
%N A352833 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.
%C A352833 A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists, so all columns k > 1 are zeros.
%C A352833 Conjecture:
%C A352833 (1) This is A064428 interleaved with A001522.
%C A352833 (2) Reversing rows gives A300788, the strict version of A300787.
%e A352833 Triangle begins:
%e A352833   0: {1,0}
%e A352833   1: {0,1}
%e A352833   2: {1,1}
%e A352833   3: {2,1}
%e A352833   4: {3,2}
%e A352833   5: {4,3}
%e A352833   6: {6,5}
%e A352833   7: {8,7}
%e A352833   8: {12,10}
%e A352833   9: {16,14}
%e A352833 For example, row n = 7 counts the following partitions:
%e A352833   (7)       (52)
%e A352833   (61)      (421)
%e A352833   (511)     (322)
%e A352833   (43)      (3211)
%e A352833   (4111)    (2221)
%e A352833   (331)     (22111)
%e A352833   (31111)   (1111111)
%e A352833   (211111)
%t A352833 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A352833 Table[Length[Select[IntegerPartitions[n],pq[#]==k&]],{n,0,15},{k,0,1}]
%Y A352833 Row sums are A000041.
%Y A352833 The version for permutations is A008290, for nonfixed points A098825.
%Y A352833 The columns appear to be A064428 and A001522.
%Y A352833 The version counting strong nonexcedances is A114088.
%Y A352833 The version for compositions is A238349, rank statistic A352512.
%Y A352833 The version for reversed partitions is A238352.
%Y A352833 Reversing rows appears to give A300788, the strict case of A300787.
%Y A352833 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A352833 A115720 and A115994 count partitions by their Durfee square.
%Y A352833 A330644 counts non-self-conjugate partitions, ranked by A352486.
%Y A352833 Cf. A000701, A219282, A257990, A350839, A352513, A352521-A352525.
%K A352833 nonn,tabf
%O A352833 0,7
%A A352833 _Gus Wiseman_, Apr 08 2022