A001277
Number of permutations of length n by rises.
Original entry on oeis.org
1, 3, 12, 56, 321, 2175, 17008, 150504, 1485465, 16170035, 192384876, 2483177808, 34554278857, 515620794591, 8212685046336, 139062777326000, 2494364438359953, 47245095998005059, 942259727190907180, 19737566982241851720, 433234326593362631601, 9943659797649140568863
Offset: 2
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A180192
Triangle read by rows: T(n,k) is the number of permutations of [n] having k fixed blocks.
Original entry on oeis.org
1, 0, 1, 1, 1, 2, 4, 0, 9, 12, 3, 44, 57, 18, 1, 265, 321, 123, 11, 1854, 2176, 888, 120, 2, 14833, 17008, 7218, 1208, 53, 133496, 150505, 65460, 12550, 860, 9, 1334961, 1485465, 657690, 137970, 12405, 309, 14684570, 16170036, 7257240, 1623440
Offset: 0
T(4,2)=3 because we have (1)4(3)2, (1)32(4), and 3(2)1(4) (the fixed blocks are shown between parentheses).
Triangle starts:
1;
0, 1;
1, 1;
2, 4, 0;
9, 12, 3;
44, 57, 18, 1;
265, 321, 123, 11;
T(2n-1,n) = d(n-1).
T(2n,n) = d(n+1) + d(n) =
A000255(n).
Sum of entries in row n is n! =
A000142(n).
-
# yields sequence in triangular form:
d[0] := 1: for n to 50 do d[n] := n*d[n-1] + (-1)^n od:
T := (n, k) -> add(binomial(j-1, k-1)*binomial(n+1-j, k)*d[n-j], j = k .. n+1-k):
for n from 0 to 11 do seq(T(n, k), k = 0..ceil((1/2)*n)) od;
-
d = Subfactorial;
T[n_, k_] := Sum[Binomial[j-1, k-1] Binomial[n-j+1, k] d[n-j] , {j, k, n-k+1}];
Table[T[n, k], {n, 0, 11}, {k, 0, Ceiling[n/2]}] // Flatten (* Jean-François Alcover, Feb 16 2021 *)
Showing 1-2 of 2 results.
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