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A352523 Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).

%I A352523 #25 Mar 21 2025 10:13:21
%S A352523 1,1,0,0,2,0,1,1,2,0,0,4,2,2,0,0,5,5,4,2,0,1,3,12,8,6,2,0,0,7,14,19,
%T A352523 14,8,2,0,0,8,21,33,32,22,10,2,0,0,9,30,54,63,54,32,12,2,0,1,6,47,80,
%U A352523 116,116,86,44,14,2,0,0,11,53,129,194,229,202,130,58,16,2,0
%N A352523 Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).
%C A352523 A nonfixed point in a composition c is an index i such that c_i != i.
%H A352523 John Tyler Rascoe, <a href="/A352523/b352523.txt">Rows n = 0..130, flattened</a>
%F A352523 G.f.: Sum_{i>=0} Product_{j=1..i} y*(x/(1-x) - x^j) + x^j. - _John Tyler Rascoe_, Mar 19 2025
%e A352523 Triangle begins:
%e A352523    1
%e A352523    1   0
%e A352523    0   2   0
%e A352523    1   1   2   0
%e A352523    0   4   2   2   0
%e A352523    0   5   5   4   2   0
%e A352523    1   3  12   8   6   2   0
%e A352523    0   7  14  19  14   8   2   0
%e A352523    0   8  21  33  32  22  10   2   0
%e A352523    0   9  30  54  63  54  32  12   2   0
%e A352523    1   6  47  80 116 116  86  44  14   2   0
%e A352523    ...
%e A352523 For example, row n = 6 counts the following compositions (empty column indicated by dot):
%e A352523   (123)  (6)   (24)    (231)    (2112)   (21111)    .
%e A352523          (15)  (33)    (312)    (2121)   (111111)
%e A352523          (42)  (51)    (411)    (3111)
%e A352523                (114)   (1113)   (11112)
%e A352523                (132)   (1122)   (11121)
%e A352523                (141)   (1311)   (11211)
%e A352523                (213)   (2211)
%e A352523                (222)   (12111)
%e A352523                (321)
%e A352523                (1131)
%e A352523                (1212)
%e A352523                (1221)
%p A352523 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A352523       add(`if`(i=j, 1, x)*b(n-j, i+1), j=1..n)))
%p A352523     end:
%p A352523 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
%p A352523 seq(T(n), n=0..12);  # _Alois P. Heinz_, Mar 19 2025
%t A352523 pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
%t A352523 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pnq[#]==k&]],{n,0,9},{k,0,n}]
%o A352523 (PARI)
%o A352523 T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h= sum(i=0, N, prod(j=1, i, y*(x/(1-x)-x^j)+x^j))); vector(N, n, my(r=Vecrev(polcoeff(h, n-1))); if(n<2, r, concat(r,[0])))}
%o A352523 T_xy(10) \\ _John Tyler Rascoe_, Mar 21 2025
%Y A352523 Column k = 0 is A010054.
%Y A352523 Row sums are A011782.
%Y A352523 The version for permutations is A098825.
%Y A352523 The corresponding rank statistic is A352513.
%Y A352523 Column k = 1 is A352520.
%Y A352523 A238349 and A238350 count comps by fixed points, first col A238351, rank stat A352512.
%Y A352523 A352486 gives the nonfixed points of A122111, counted by A330644.
%Y A352523 A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
%Y A352523 A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.
%Y A352523 A352524 counts comps by strong excedances, first col A008930, stat A352516.
%Y A352523 A352525 counts comps by weak excedances, first col A177510, stat A352517.
%Y A352523 Cf. A000700, A064428, A088218, A114088, A115994, A123125, A173018, A238352.
%K A352523 nonn,tabl
%O A352523 0,5
%A A352523 _Gus Wiseman_, Mar 26 2022