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

Author

Edwin Hermann, Jun 09 2025

Keywords

• nonn

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.

Examples

			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 .
  . . . . .    . . . . .    . . . . .
		

Crossrefs

Cf. A035290, A294240.

Programs

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

Extensions

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