cp's OEIS Frontend

A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

%I A352828 #14 Jul 22 2024 16:45:17
%S A352828 1,0,1,2,2,2,2,3,4,6,8,10,12,14,16,19,22,26,32,38,46,56,66,78,92,106,
%T A352828 123,142,162,186,214,244,280,322,368,422,484,552,630,718,815,924,1046,
%U A352828 1180,1330,1498,1682,1888,2118,2372,2656,2972,3322,3712,4146,4626
%N A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.
%H A352828 Alois P. Heinz, <a href="/A352828/b352828.txt">Table of n, a(n) for n = 0..5000</a>
%F A352828 G.f.: Sum_{n>=0} q^(n*(3*n+1)/2)*Product_{k=1..n} (1+q^k)/(1-q^k). - _Jeremy Lovejoy_, Sep 26 2022
%e A352828 The a(0) = 1 through a(12) = 12 partitions (A-C = 10..12; empty column indicated by dot; 0 is the empty partition):
%e A352828    0  .  2  3    4    5    6    7    8     9     A      B      C
%e A352828             21   31   41   51   43   53    54    64     65     75
%e A352828                                 61   71    63    73     74     84
%e A352828                                      431   81    91     83     93
%e A352828                                            432   532    A1     B1
%e A352828                                            531   541    542    642
%e A352828                                                  631    632    651
%e A352828                                                  4321   641    732
%e A352828                                                         731    741
%e A352828                                                         5321   831
%e A352828                                                                5421
%e A352828                                                                6321
%t A352828 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A352828 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]==0&]],{n,0,30}]
%Y A352828 The version for permutations is A000166, complement A002467.
%Y A352828 The reverse version is A025147, complement A238395, non-strict A238394.
%Y A352828 The non-strict version is A064428 (unproved, ranked by A352826 or A352873).
%Y A352828 The version for compositions is A238351, complement A352875.
%Y A352828 The complement is A352829, non-strict A001522 (unproved, ranked by A352827 or A352874).
%Y A352828 A000041 counts partitions, strict A000009.
%Y A352828 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A352828 A008290 counts permutations by fixed points, unfixed A098825.
%Y A352828 A115720 and A115994 count partitions by their Durfee square.
%Y A352828 A238349 counts compositions by fixed points, complement A352523.
%Y A352828 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A352828 A352833 counts partitions by fixed points.
%Y A352828 Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352824, A352825, A352830, A352872.
%K A352828 nonn
%O A352828 0,4
%A A352828 _Gus Wiseman_, May 15 2022