A392945 Irregular triangular array T(n,k) read by rows. Row n gives the number of permutations of {1,2,...,n} whose descent set is contained in S for each subset S of {1,2,...,n-1} where the subsets are ordered first by size and then lexicographically (Cf. A082185).

1, 1, 1, 2, 1, 3, 3, 6, 1, 4, 6, 4, 12, 12, 12, 24, 1, 5, 10, 10, 5, 20, 30, 20, 30, 30, 20, 60, 60, 60, 60, 120, 1, 6, 15, 20, 15, 6, 30, 60, 60, 30, 60, 90, 60, 60, 60, 30, 120, 180, 120, 180, 180, 120, 180, 180, 180, 120, 360, 360, 360, 360, 360, 720

Offset

0,4

Comments

Let S be the k-th subset of [n-1]. T(n,k) is the number of maximal chains in the S-rank selected poset of the Boolean lattice on [n]. See Stanley reference below.

References

Richard Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Section 3.13.

Links

Alois P. Heinz, Rows n = 0..15, flattened

Formula

Let S = {s_1<s_2<...<s_k}. T(n,S) = multinomial(n;s_1,s_2-s_1,s_3-s_2,...,n-s_k). See Stanley reference above and Example 2.2.4.

T(n,S) = Sum_A335845(n,T) where the sum is over every subset T of S.

Example

Triangle begins:
  1;
  1;
  1, 2;
  1, 3,  3,  6;
  1, 4,  6,  4, 12, 12, 12, 24;
  1, 5, 10, 10,  5, 20, 30, 20, 30, 30, 20, 60, 60, 60, 60, 120;
  ...

Maple

T:= proc(n) option remember; uses combinat; `if`(n=0, 1, map(l->
     `if`(l=[], 1, multinomial(n, l[1], l[i]-l[i-1]$i=2..nops(l),
      n-l[-1])), [seq(choose([$1..n-1], i)[], i=0..n-1)])[])
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 27 2026

Mathematica

descentset[permutation_] := Flatten[Position[Sign[Differences[permutation]], -1]]; alpha[n_, subset_] := Select[Permutations[Range[n]], MemberQ[Subsets[subset], descentset[#]] &] // Length; flagfvectorboolean[n_] := Map[alpha[n, #] &, Subsets[Range[n - 1]]]; Table[flagfvectorboolean[n], {n, 0, 6}]

Crossrefs

Cf. A000142, A335845, A082185, A000670 (row sums).

Sequence in context: A132883 A132888 A213934 * A124774 A056610 A341450

Adjacent sequences: A392942 A392943 A392944 * A392946 A392947 A392948

Keyword

nonn,look,tabf

Author

Geoffrey Critzer, Jan 27 2026

Status

approved