cp's OEIS Frontend

A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

%I A353318 #5 May 21 2022 14:50:29
%S A353318 1,1,1,1,2,1,4,1,6,1,9,1,1,12,2,1,16,5,1,20,9,1,25,16,1,30,25,1,36,39,
%T A353318 1,1,42,56,2,1,49,80,5,1,56,109,10,1,64,147,19,1,72,192,32,1,81,249,
%U A353318 54,1,90,315,84,1,100,396,129,1,1,110,489,190,2,1,121,600,275,5
%N A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.
%e A353318 Triangle begins:
%e A353318    1
%e A353318    1   1
%e A353318    1   2
%e A353318    1   4
%e A353318    1   6
%e A353318    1   9   1
%e A353318    1  12   2
%e A353318    1  16   5
%e A353318    1  20   9
%e A353318    1  25  16
%e A353318    1  30  25
%e A353318    1  36  39   1
%e A353318    1  42  56   2
%e A353318    1  49  80   5
%e A353318    1  56 109  10
%e A353318 For example, row n = 7 counts the following partitions:
%e A353318   (1111111)  (7)       (43)
%e A353318              (52)      (331)
%e A353318              (61)
%e A353318              (322)
%e A353318              (421)
%e A353318              (511)
%e A353318              (2221)
%e A353318              (3211)
%e A353318              (4111)
%e A353318              (22111)
%e A353318              (31111)
%e A353318              (211111)
%t A353318 partsabove[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t A353318 DeleteCases[Table[Length[Select[IntegerPartitions[n],partsabove[#]==k&]],{n,1,15},{k,0,n-1}],0,2]
%Y A353318 Row sums are A000041.
%Y A353318 Row lengths are A000194, reversed A003056.
%Y A353318 Column k = 1 is A002620, reversed A238875.
%Y A353318 Column k = 2 is A097701.
%Y A353318 The version for permutations is A008292, opposite A123125.
%Y A353318 The weak version is A115720/A115994, rank statistic A257990.
%Y A353318 The version for compositions is A352524, weak A352525.
%Y A353318 The version for reversed partitions is A353319.
%Y A353318 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A353318 A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
%Y A353318 A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
%Y A353318 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A353318 Cf. A000701, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.
%K A353318 nonn,tabf
%O A353318 1,5
%A A353318 _Gus Wiseman_, May 21 2022