cp's OEIS Frontend

A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

%I A352829 #14 Oct 16 2022 23:32:49
%S A352829 0,1,0,0,0,1,2,2,2,2,2,2,3,4,6,8,10,12,14,16,18,20,23,26,30,36,42,50,
%T A352829 60,70,82,96,110,126,144,163,184,208,234,264,298,336,380,430,486,550,
%U A352829 622,702,792,892,1002,1125,1260,1408,1572,1752,1950,2168,2408,2672
%N A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.
%F A352829 G.f.: Sum_{n>=1} q^(n*(3*n-1)/2)*Product_{k=1..n-1} (1+q^k)/(1-q^k). - _Jeremy Lovejoy_, Sep 26 2022
%e A352829 The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
%e A352829   (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
%e A352829   (821)  (543)  (643)   (653)   (753)    (763)    (863)
%e A352829          (921)  (A21)   (743)   (843)    (853)    (953)
%e A352829                 (5431)  (B21)   (C21)    (943)    (A43)
%e A352829                         (5432)  (6432)   (D21)    (E21)
%e A352829                         (6431)  (6531)   (6532)   (7532)
%e A352829                                 (7431)   (7432)   (7631)
%e A352829                                 (54321)  (7531)   (8432)
%e A352829                                          (8431)   (8531)
%e A352829                                          (64321)  (9431)
%e A352829                                                   (65321)
%e A352829                                                   (74321)
%t A352829 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A352829 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]>0&]],{n,0,30}]
%Y A352829 The non-strict version is A001522 (unproved, ranked by A352827 or A352874).
%Y A352829 The version for permutations is A002467, complement A000166.
%Y A352829 The reverse version is A096765 (or A025147 shifted right once).
%Y A352829 The non-strict reverse version is A238395, ranked by A352872.
%Y A352829 The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).
%Y A352829 The version for compositions is A352875, complement A238351.
%Y A352829 A000041 counts partitions, strict A000009.
%Y A352829 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A352829 A008290 counts permutations by fixed points, unfixed A098825.
%Y A352829 A115720 and A115994 count partitions by their Durfee square.
%Y A352829 A238349 counts compositions by fixed points, complement A352523.
%Y A352829 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A352829 A238394 counts reversed partitions without a fixed point, ranked by A352830.
%Y A352829 A352833 counts partitions by fixed points.
%Y A352829 Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.
%K A352829 nonn
%O A352829 0,7
%A A352829 _Gus Wiseman_, May 15 2022