Abramson, M. and Moser, W. O. J. "Permutations without Rising or Falling -Sequences." Ann. Math. Statist.38, 1245-1254, 1967.André, D. "Mémoir sur les couples actifs de permutations." Mem. della Pontificia Acad. Romana dei Nuovo Lincei23, 189-223, 1906.Carlitz, L. "Note on a Paper of Shanks." Amer. Math. Monthly59, 239-241, 1952.Carlitz, L. "Eulerian Numbers and Polynomials." Math. Mag.32, 247-260, 1959.Carlitz, L. "Eulerian Numbers and Polynomials of Higher Order." Duke Math. J.27, 401-423, 1960.Carlitz, L. "A Note on the Eulerian Numbers." Arch. Math.14, 383-390, 1963.Carlitz, L. and Riordan, J. "Congruences for Eulerian Numbers." Duke Math. J.20, 339-343, 1953.Carlitz, L.; Roselle, D. P.; and Scoville, R. "Permutations and Sequences with Repetitions by Number of Increase." J. Combin. Th.1, 350-374, 1966.Cesàro, E. "Dérivées des fonctions de fonctions." Nouv. Ann.5, 305-327, 1886.Comtet, L. "Permutations by Number of Rises; Eulerian Numbers." §6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. 多德雷赫特，荷兰: Reidel, pp. 51 and 240-246, 1974.David, F. N.; Kendall, M. G.; and Barton, D. E. Symmetric Function and Allied Tables. 剑桥，英国: 剑桥大学出版社, p. 260, 1966.Dillon, J. F.; Roselle, D. P. "Eulerian Numbers of Higher Order." Duke Math. J.35, 247-256, 1968.Foata, D. and Schützenberger, M.-P. Théorie géométrique des polynômes Eulériens. 柏林: Springer-Verlag, 1970.Frobenius, F. G. "Ueber die Bernoullischen Zahlen und die Eulerischen Polynome." Sitzungsber. Preuss. Akad. Wiss., pp. 808-847, 1910.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Eulerian Numbers." §6.2 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. 雷丁，马萨诸塞州: Addison-Wesley, pp. 267-272, 1994.Kimber, A. C. "Eulerian Numbers." Supplement to Encyclopedia of Statistical Sciences. (Ed. S. Kotz, N. L. Johnson, and C. B. Read). 纽约: Wiley, pp. 59-60, 1989.Poussin, F. "Sur une propriété arithmétique de certains polynomes associés aux nombres d'Euler." C. R. Acad. Sci. Paris Sér. A-B266, A392-A393, 1968.Salama, I. A. and Kupper, L. L. "A Geometric Interpretation for the Eulerian Numbers." Amer. Math. Monthly93, 51-52, 1986.Schrutka, L. "Eine neue Einleitung der Permutationen." Math. Ann.118, 246-250, 1941.Shanks, E. B. "Iterated Sums of Powers of the Binomial Coefficients." Amer. Math. Monthly58, 404-407, 1951.Sloane, N. J. A. 序列 A000295/M3416, A000460/M4795, A000498/M5188, 和 A008292，出自 "The On-Line Encyclopedia of Integer Sequences."Tomić, M. "Sur une nouvelle classe de polynômes de la théorie des fonctions spéciales." Publ. Fac. Elect. U. Belgrade, No. 38, 1960.Toscano, L. "Su due sviluppi della potenza di un binomio, -coefficienti di Eulero." Bull. S. M. Calabrese16, 1-8, 1965.
给出了集合 
中恰好有 
个 排列升序 的排列数（Graham et al. 1994, p. 267）。请注意，Comtet (1974) 使用了略有不同的欧拉数定义，他将欧拉数 
(有时也表示为 
) 定义为长度为 
的 排列游程 的数量，因此 
。
是 多对数函数。因此，
由 
在以下表达式中的系数给出
具有 指数生成函数
满足 递推关系
, 
, 
, ...
排列成三角形，得到 欧拉数三角形
-Sequences." Ann. Math. Statist. 38, 1245-1254, 1967.André, D. "Mémoir sur les couples actifs de permutations." Mem. della Pontificia Acad. Romana dei Nuovo Lincei 23, 189-223, 1906.Carlitz, L. "Note on a Paper of Shanks." Amer. Math. Monthly 59, 239-241, 1952.Carlitz, L. "Eulerian Numbers and Polynomials." Math. Mag. 32, 247-260, 1959.Carlitz, L. "Eulerian Numbers and Polynomials of Higher Order." Duke Math. J. 27, 401-423, 1960.Carlitz, L. "A Note on the Eulerian Numbers." Arch. Math. 14, 383-390, 1963.Carlitz, L. and Riordan, J. "Congruences for Eulerian Numbers." Duke Math. J. 20, 339-343, 1953.Carlitz, L.; Roselle, D. P.; and Scoville, R. "Permutations and Sequences with Repetitions by Number of Increase." J. Combin. Th. 1, 350-374, 1966.Cesàro, E. "Dérivées des fonctions de fonctions." Nouv. Ann. 5, 305-327, 1886.Comtet, L. "Permutations by Number of Rises; Eulerian Numbers." §6.5 in 
-coefficienti di Eulero." Bull. S. M. Calabrese 16, 1-8, 1965.