A384759 Number of legal arrangements in pawn-only chess on an n X n board where no pieces have been taken and no piece attacks another piece.

0, 3, 2031, 728174, 247646098, 91880342535, 38818192375310, 18907485764545412, 10626953883068264472, 6866760686250915376779, 5073038373153476636807709, 4259014676256866422905669602, 4038463837000965678262091166880, 4299625631242136963071149921577615, 5111407212497576694797045579672852791

Offset

4,2

Comments

The number of ways of arranging n pawns of each color on an n X n board such that no pawn threatens another, each file contains one pawn of each color, none of the pawns are passed pawns, and each pawn is placed between row 2 and row n-1 inclusive.

There is no requirement that the arrangements counted here can actually be achieved via a sequence of legal chess moves.

Links

Andrew Howroyd, Table of n, a(n) for n = 4..40

Example

The a(5) = 3 positions are:
  . . . . .    . . . . .    . . . . .
  b b b b b    . b . b .    b . b . b
  . . . . .    b w b w b    w b w b w
  w w w w w    w . w . w    . w . w .
  . . . . .    . . . . .    . . . . .

Prog

(PARI)
MkTfrMtx(n)={my(m=binomial(n, 2), M=matrix(m, m)); for(i=1, n-1, for(j=i+1, n, for(p=1, n-1, for(q=p+1, n, if(q<>i+1&&j<>p+1, M[binomial(n-i, 2)+(j-i), binomial(n-p, 2)+(q-p)]=1) )))); M}
a(n)={my(M=MkTfrMtx(n-2)); vecsum(M^(n-1)*vectorv(#M, i, 1))} \\ Andrew Howroyd, Jun 15 2025

Crossrefs

Cf. A035290, A294240.

Sequence in context: A236481 A307928 A193149 * A024047 A102987 A004814

Adjacent sequences: A384756 A384757 A384758 * A384760 A384761 A384762

Keyword

nonn

Author

Edwin Hermann, Jun 09 2025

Extensions

a(9) onwards from Andrew Howroyd, Jun 15 2025

Status

approved