A390925 a(n) is the number of 3 X 3 matrices with elements 1..n where at least one row and at least one column is strictly increasing.

0, 0, 0, 243, 8608, 100324, 671092, 3185367, 11947344, 37713112, 104318488, 259918219, 594918544, 1269236540, 2552136236, 4878573167, 8926725856, 15722204656, 26775305456, 44258617955, 71233305536, 111933446228, 172118961828, 259508864967, 384307821744, 559840360520

Offset

0,4

Links

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = n*(n - 1)*(n - 2)*(8141*n^6 - 14637*n^5 - 11887*n^4 + 12765*n^3 + 8750*n^2 - 540*n + 144)/45360. - Jason Yuen, Nov 25 2025

Mathematica

a[n_, d_] := a[n, d] = (AnyTrue[#, OrderedQ[#, Less] &] && AnyTrue[Transpose[#], OrderedQ[#, Less] &] &) /@ Tuples[Range[1, n], {d, d}] // Boole // Total;
Table[a[n, 3], {n, 0, 7}]

Prog

(PARI) a(n)={res=0; forvec(x=vector(9, i, [1, n]), m=matrix(3, 3); m[1, ]=x[1..3]; m[2, ]=x[4..6]; m[3, ]=x[7..9]; res+=isvalidmatrix(m)); res}
isvalidmatrix(m)={rowsok=0; for(i=1, 3, if(m[i, 1] < m[i, 2] && m[i, 2] < m[i, 3], rowsok=1; break)); if(rowsok==0, return(0)); columnsok=0; for(i=1, 3, if(m[1, i] < m[2, i] && m[2, i] < m[3, i], columnsok = 1; break)); if(columnsok==0, return(0)); return(1)} \\ David A. Corneth, Nov 24 2025
(Python)
def A390925(n): return (n*(n*(n*(n*(n*(n*(n*(n*(8141*n-39060)+48306)+19152)-53319)-1260)+19264)-1512)+288)>>4)//2835 # Chai Wah Wu, Dec 02 2025

Crossrefs

Cf. A390922 (2 X 2).

Cf. A206808 (2 X 2, row only), A390904 (3 X 3, row only).

Sequence in context: A224015 A059860 A268975 * A223207 A224313 A224377

Adjacent sequences: A390922 A390923 A390924 * A390926 A390927 A390928

Keyword

nonn,easy

Author

Robert P. P. McKone, Nov 24 2025

Extensions

a(8)-a(17) from Jason Yuen, Nov 25 2025

a(18)-a(20) from Sean A. Irvine, Dec 01 2025

a(21)-a(25) from Jason Yuen, Dec 01 2025

Status

approved