A347930 3-Springer numbers.

1, 1, 3, 16, 88, 625, 5527, 55760, 640540, 8329326, 120212331, 1905939913, 32987637967, 618591571085, 12489644875037, 270193806214360, 6235154917414954, 152875655211527878, 3968729594485785289, 108754865309750398187, 3137052120203959610759

Offset: 2

Author

Alejandro H. Morales, Sep 19 2021

Keywords

• nonn

Comments

a(n) is also the volume of a certain flow polytope.

Links

• Arvind Ayyer, Matthieu Josuat-Vergès, and Sanjay Ramassamy, Extensions of partial cyclic orders and consecutive coordinate polytopes, Ann. H. Lebesgue, 3 (2020), 275-297.
• R. S. Gonzalez D'Leon, A. H. Morales, C. R. H. Hanusa, and M. Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
• S. Ramassamy, Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons, Electron. J. Combin., 25 (2018), #P1.66.

Crossrefs

Cf. A001586, A008282, A000111.

Programs

Maple

wcomps:=proc(n,k)
       option remember;
local ocomps,ncomps,i;
ocomps:=combinat:-composition(n+k,k);
ncomps:={};
for i from 1 to nops(ocomps) do
   ncomps:=ncomps union{[seq(ocomps[i][j]-1,j=1..k)]};
end do;
return [op(ncomps)];
end proc:
b:=proc(s) option remember;
   local k;
   k := nops(s);
   if s = [seq(0,i=1..k)] then
      return(1);
   elif s[1]>0 then
      return(add(b([s[2]+j,op(s[3..k]),s[1]-j-1]),j=0..s[1]-1));
   else
      return(0);
   end if;
end proc:a:=proc(n)   local N,S:   N := n-2;   S := wcomps(N,3);   return add(combinat:-multinomial(N,op(s))*b(s), s in S);end proc:seq(a(n),n=2..10);

Formula

a(n) = Sum_{(x,y,z), x+y+z=n-2} ((n-2)!/(x!*y!*z!))*b(x,y,z), where b(x,y,z) are the 3-Entringer numbers defined by Ramassamy.