A189284 -id:A189284

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A189285

Number of permutations p of 1,2,...,n satisfying p(i+6)-p(i)<>6 for all 1<=i<=n-6.

+10
2

1, 1, 2, 6, 24, 120, 720, 4920, 37488, 319644, 3033264, 31784280, 364902480, 4538652840, 61102571376, 885045657564, 13722397569072, 226742901078120, 3977354871110160, 73816786920489720, 1444940702597713008, 29750236302549282948

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[6,6] on an n X n chessboard.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..24 (Updated Jan 19 2019)

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 644.

Vaclav Kotesovec, Mathematica program for this sequence

George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.

FORMULA

Asymptotics (V. Kotesovec, Mar 2011): a(n)/n! ~ (1 + 11/n + 30/n^2)/e.

Generally (for this sequence is d=6): 1/e*(1+(2d-1)/n+d*(d-1)/n^2).

CROSSREFS

Cf. A000255, A189281, A189282, A189283, A189284, A189271.

KEYWORD

nonn,hard

AUTHOR

Vaclav Kotesovec, Apr 19 2011

EXTENSIONS

Terms a(23)-a(24) from Vaclav Kotesovec, Apr 21 2012

STATUS

approved

A189847

Number of ways to place n nonattacking composite pieces rook + semi-rider[5,5] on an n X n chessboard.

+10
1

1, 2, 6, 24, 120, 696, 4572, 34260, 290328, 2751480, 28426056, 318900264, 3874868280, 50813711808, 716309557440, 10721493269568

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+5k)-p(j)<>5k for all j>=1, k>=1, j+5k<=n

For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

LINKS

Table of n, a(n) for n=1..16.

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Wikipedia, Fairy chess piece