cp's OEIS Frontend

A352875 Number of integer compositions y of n with a fixed point y(i) = i.

%I A352875 #10 Jan 02 2023 21:55:08
%S A352875 0,1,1,2,5,10,21,42,86,174,351,708,1424,2861,5743,11520,23092,46269,
%T A352875 92673,185562,371469,743491,1487870,2977164,5956616,11916910,23839736,
%U A352875 47688994,95393322,190811346,381662507,763389209,1526881959,3053930971,6108131542,12216698288
%N A352875 Number of integer compositions y of n with a fixed point y(i) = i.
%H A352875 Andrew Howroyd, <a href="/A352875/b352875.txt">Table of n, a(n) for n = 0..1000</a>
%F A352875 a(n) = 2^(n-1) - A238351(n) for n >= 1. - _Andrew Howroyd_, Jan 02 2023
%e A352875 The a(0) = 0 through a(5) = 10 compositions (empty column indicated by dot):
%e A352875   .  (1)  (11)  (12)   (13)    (14)
%e A352875                 (111)  (22)    (32)
%e A352875                        (112)   (113)
%e A352875                        (121)   (122)
%e A352875                        (1111)  (131)
%e A352875                                (221)
%e A352875                                (1112)
%e A352875                                (1121)
%e A352875                                (1211)
%e A352875                                (11111)
%t A352875 pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t A352875 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pq[#]>0&]],{n,0,15}]
%o A352875 (PARI)
%o A352875 S(v,u,c)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
%o A352875 seq(n)={my(v=vector(1+n), s=vector(#v, i, 2^(i-2))); v[1]=1; s[1]=0; for(i=1, n, v=S(v, vector(n, j, if(j==i,'x,1)), O(x)); s-=apply(p->polcoef(p,0), v)); s} \\ _Andrew Howroyd_, Jan 02 2023
%Y A352875 The version for partitions is A001522, ranked by A352827 (unproved).
%Y A352875 The version for permutations is A002467, complement A000166.
%Y A352875 The complement for partitions is A064428, ranked by A352826 (unproved).
%Y A352875 This is the sum of latter columns of A238349, nonfixed A352523.
%Y A352875 The complement is counted by A238351.
%Y A352875 The complement for reversed partitions is A238394, ranked by A352830.
%Y A352875 The version for reversed partitions is A238395, ranked by A352872.
%Y A352875 The case of just one fixed point is A240736.
%Y A352875 A008290 counts permutations by fixed points, nonfixed A098825.
%Y A352875 A011782 counts compositions.
%Y A352875 A115720 and A115994 count partitions by Durfee square.
%Y A352875 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A352875 A352512 counts fixed points in standard compositions, nonfixed A352513.
%Y A352875 A352521 = comps by subdiags, first col A219282, rank stat A352514.
%Y A352875 A352522 = comps by weak subdiags, first col A238874, rank stat A352515.
%Y A352875 A352524 = comps by superdiags, first col A008930, rank stat A352516.
%Y A352875 A352525 = comps by weak superdiags, col k=1 A177510, rank stat A352517.
%Y A352875 A352833 counts partitions by fixed points.
%Y A352875 Cf. A006918, A008292, A088218, A114088, A123125, A188674, A257990.
%K A352875 nonn
%O A352875 0,4
%A A352875 _Gus Wiseman_, May 15 2022
%E A352875 Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2023