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See A049459 for a signed version of the array. The unsigned 4-Stirling numbers of the first kind count the permutations of the set {1,2,...,n} into k disjoint cycles, with the restriction that the elements 1, 2, 3 and 4 belong to distinct cycles. This is the case r = 4 of the unsigned r-Stirling numbers of the first kind. For other cases see abs(A008275) (r = 1), A143491 (r = 2) and A143492 (r = 3). See A143496 for the corresponding triangle of 4-Stirling numbers of the second kind.

The theory of r-Stirling numbers of both kinds is developed in [Broder]. For details of the related 4-Lah numbers see A143499.

With offset n=0 and k=0, this is the Sheffer triangle (1/(1-x)^4,-log(1-x)) (in the umbral notation of S. Roman's book this would be called Sheffer for (exp(-4*t),1-exp(-t))). See the e.g.f given below. Compare also with the e.g.f. for the signed version A049459. - Wolfdieter Lang, Oct 10 2011

With offset n=0 and k=0: triangle T(n,k), read by rows, given by (4,1,5,2,6,3,7,4,8,5,9,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 31 2011

This is the case r = 4 of the r-Eulerian numbers, denoted by A(r;n,k), defined as follows. Let [n] denote the ordered set {1,2,...,n} and let r be a nonnegative integer. Let Permute(n,n-r) denote the set of injective maps p:[n-r] -> [n], which we think of as permutations of n numbers taken n-r at a time. Clearly, |Permute(n,n-r)| = n!/r!. We say that the permutation p has an excedance at position i, 1 <= i <= n-r, if p(i) > i. Then the r-Eulerian number A(r;n,k) is defined as the number of permutations in Permute(n,n-r) with k excedances. Thus the 4-Eulerian numbers are the number of permutations in Permute(n,n-4) with k excedances. For other cases see A008292 (r = 0 and r = 1), A144696 (r = 2), A144697 (r = 3) and A144699 (r = 5).

An alternative interpretation of the current array due to [Strosser] involves the 4-excedance statistic of a permutation (see also [Foata & Schutzenberger, Chapter 4, Section 3]). We define a permutation p in Permute(n,n-4) to have a 4-excedance at position i (1 <= i <= n-4) if p(i) >= i + 4.

Given a permutation p in Permute(n,n-4), define ~p to be the permutation in Permute(n,n-4) that takes i to n+1 - p(n-i-3). The map ~ is a bijection of Permute(n,n-4) with the property that if p has (resp. does not have) an excedance in position i then ~p does not have (resp. has) a 4-excedance at position n-i-3. Hence ~ gives a bijection between the set of permutations with k excedances and the set of permutations with (n-k) 4-excedances. Thus reading the rows of this array in reverse order gives a triangle whose entries are the number of permutations in Permute(n,n-4) with k 4-excedances.

Example: Represent a permutation p:[n-4] -> [n] in Permute(n,n-4) by its image vector (p(1),...,p(n-4)). In Permute(10,6) the permutation p = (1,2,4,10,3,6) does not have an excedance in the first two positions (i = 1 and 2) or in the final two positions (i = 5 and 6). The permutation ~p = (5,8,1,7,9,10) has 4-excedances only in the first two positions and the final two positions.

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

If in the partition array M31hat(5):=A144890 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(5). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first columns are A001720(n+3)=(n+3)!/4!, A144893, A144894,...