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A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.

%I A353319 #13 May 21 2022 14:54:16
%S A353319 1,1,1,2,1,2,3,4,2,1,5,4,2,7,6,2,10,6,6,15,7,7,1,18,14,7,3,26,15,11,4,
%T A353319 35,17,19,6,47,24,19,11,61,33,22,18,1,80,44,28,20,4,103,54,42,25,7,
%U A353319 138,60,57,31,11,175,85,58,52,15,224,112,66,64,24
%N A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.
%e A353319 Triangle begins:
%e A353319    1
%e A353319    1  1
%e A353319    2  1
%e A353319    2  3
%e A353319    4  2  1
%e A353319    5  4  2
%e A353319    7  6  2
%e A353319   10  6  6
%e A353319   15  7  7  1
%e A353319   18 14  7  3
%e A353319   26 15 11  4
%e A353319   35 17 19  6
%e A353319   47 24 19 11
%e A353319   61 33 22 18  1
%e A353319   80 44 28 20  4
%e A353319 For example, row n = 9 counts the following reversed partitions:
%e A353319   (1134)       (9)     (27)   (234)
%e A353319   (1224)       (18)    (36)
%e A353319   (1233)       (117)   (45)
%e A353319   (11115)      (126)   (135)
%e A353319   (11124)      (1116)  (144)
%e A353319   (11133)      (1125)  (225)
%e A353319   (11223)      (2223)  (333)
%e A353319   (12222)
%e A353319   (111114)
%e A353319   (111123)
%e A353319   (111222)
%e A353319   (1111113)
%e A353319   (1111122)
%e A353319   (11111112)
%e A353319   (111111111)
%t A353319 partsabove[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t A353319 DeleteCases[Table[Length[Select[Reverse/@IntegerPartitions[n], partsabove[#]==k&]],{n,1,30},{k,0,n-1}],0,2]
%Y A353319 Row sums are A000041.
%Y A353319 Row lengths are A003056.
%Y A353319 The version for permutations is A008292, opposite A123125.
%Y A353319 The weak unreversed version is A115720/A115994, rank statistic A257990.
%Y A353319 For fixed points instead of excedances we have A238352, rank stat A352822.
%Y A353319 Column k = 0 is A238875.
%Y A353319 The version for compositions is A352524, weak A352525.
%Y A353319 The version for unreversed partitions is A353318.
%Y A353319 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A353319 A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
%Y A353319 A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
%Y A353319 Cf. A000701, A002620, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.
%K A353319 nonn,tabf
%O A353319 1,4
%A A353319 _Gus Wiseman_, May 21 2022