This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373418 #40 Mar 30 2025 09:53:02
%S A373418 0,0,0,0,0,1,0,0,3,2,0,0,6,8,6,0,0,10,20,30,24,0,0,15,40,90,144,135,0,
%T A373418 0,21,70,210,504,945,930,0,0,28,112,420,1344,3780,7440,7420,0,0,36,
%U A373418 168,756,3024,11340,33480,66780,66752,0,0,45,240,1260,6048,28350,111600,333900,667520,667485
%N A373418 Triangle read by rows: T(n,k) is the number of permutations in symmetric group S_n with (n-k) fixed points and an odd number of non-fixed point cycles. Equivalent to the number of cycles of n items with cycle type defined by non-unity partitions of k <= n that contain an odd number of parts.
%C A373418 a(n) + A343417(n) = A098825(n) = partial derangement "rencontres" triangle.
%C A373418 a(n) - A343417(n) = (k-1) * binomial(n,k) = A127717(n-1,k-1).
%C A373418 Difference of 2nd and 1st leading diagonals (n > 0):
%C A373418 T(n,n-1) - T(n,n) = 0,-1,1,2,6,9,15,20,28,35,45,54,...
%C A373418 = (0-1) + (2+1) + (4+3) + (6+5) + (8+7) + (10+9) + ...
%C A373418 Cf. A084265(n) with 2 terms 0,-1 prepended (moving its offset from 0 to -2).
%F A373418 T(n,k) = (n!/(n-k)!/2) * ((Sum_{j=0..k} (-1)^j/j!) + (k-1)/k!). Cf. Sum_{j=0..k} (-1)^j/j! = A053557(k) / A053556(k).
%e A373418 Triangle begins:
%e A373418 n: {k<=n}
%e A373418 0: {0}
%e A373418 1: {0, 0}
%e A373418 2: {0, 0, 1}
%e A373418 3: {0, 0, 3, 2}
%e A373418 4: {0, 0, 6, 8, 6}
%e A373418 5: {0, 0, 10, 20, 30, 24}
%e A373418 6: {0, 0, 15, 40, 90, 144, 135}
%e A373418 7: {0, 0, 21, 70, 210, 504, 945, 930}
%e A373418 8: {0, 0, 28, 112, 420, 1344, 3780, 7440, 7420}
%e A373418 9: {0, 0, 36, 168, 756, 3024, 11340, 33480, 66780, 66752}
%e A373418 10: {0, 0, 45, 240, 1260, 6048, 28350, 111600, 333900, 667520, 667485}
%e A373418 T(n,0) = 0 because the sole permutation in S_n with n fixed points, namely the identity permutation, has 0 non-fixed point cycles, not an odd number.
%e A373418 T(n,1) = 0 because there are no permutations in S_n with (n-1) fixed points.
%e A373418 Example:
%e A373418 T(3,3) = 2 since S_3 contains 3 permutations with 0 fixed points and an odd number of non-fixed point cycles, namely the derangements (123) and (132).
%e A373418 Worked Example:
%e A373418 T(7,6) = 945 permutations in S_7 with 1 fixed point and an odd number of non-fixed point cycles;
%e A373418 T(7,6) = 945 possible 6- and (2,2,2)-cycles of 7 items.
%e A373418 N(n,y) = possible y-cycles of n items;
%e A373418 N(n,y) = (n!/(n-k)!) / (M(y) * s(y)).
%e A373418 y = partition of k<=n with q parts = (p_1, p_2, ..., p_i, ..., p_q) such that k = Sum_{i=1..q} p_i.
%e A373418 Or:
%e A373418 y = partition of k<=n with d distinct parts, each with multiplicity m_j = (y_1^m_1, y_2^m_2, ..., y_j^m_j, ..., y_d^m_d) such that k = Sum_{j=1..d} m_j*y_j.
%e A373418 M(y) = Product_{i=1..q} p_i = Product_{j=1..d} y_j^m_j.
%e A373418 s(y) = Product_{j=1..d} m_j! ("symmetry of repeated parts").
%e A373418 Note: (n!/(n-k)!) / s(y) = multinomial(n, {m_j}).
%e A373418 Therefore:
%e A373418 T(7,6) = N(7,y=(6)) + N(7,y=(2^3))
%e A373418 = (7!/6) + (7!/(2^3)/3!)
%e A373418 = 7! * (1/6 + 1/48)
%e A373418 = 5040 * (3/16);
%e A373418 T(7,6) = 945.
%p A373418 b:= proc(n, t) option remember; `if`(n=0, t, add(expand(`if`(j>1, x^j, 1)*
%p A373418 b(n-j, irem(t+signum(j-1), 2)))*binomial(n-1, j-1)*(j-1)!, j=1..n))
%p A373418 end:
%p A373418 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
%p A373418 seq(T(n), n=0..10);
%t A373418 Table[Table[n!/(n-k)!/2 * (Sum[(-1)^j/j!, {j, 0, k}] - ((k - 1)/k!)),{k,1,n}], {n,1,10}]
%Y A373418 Cf. A373417 (even case), A373340 (row sums), A216779 (main diagonal).
%K A373418 nonn,tabl
%O A373418 0,9
%A A373418 _Julian Hatfield Iacoponi_, Jun 04 2024